Algebraic and Geometric Combinatorics (Mathematics Studies)

ALGEBRAIC AND GEOMETRIC COMBINATORICS

General Editor

Peter L. HAMMER. Unive&ity of Waterloo, Ont.. Canada Adcisory Editors

C . BERGE. Universite de Paris. France M . A . HARRISON. University of California. Berkeley. CA. U.S.A. V KLEE. University of Washington. Seattle, WA. U.S.A. J . H . VAN LINT, California Institute of Technology. Pasadena, CA, U.S.A. G.-C. ROTA. Massachusetts Institute of Technology, Cambridge, MA. U.S.A.

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NORTH-HOLLAND MATHEMATICS STUDIES

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Annals of Discrete Mathematics(15) General Editor: Peter

L. Hammer

University of Waterloo, Ont., Canada

Algebraic and Geometric Editor:

Eric MENDELSOHN Department of Mathematics University of Toronto Toronto, Ontario. Canada

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Algebraic and geometric combinatorics. (Annals of discrete mathematics; 15) (North-Holland mathematics studies; 65) Bibliography: p. 1. Combinatorid analysis-- Addresses, essays, lectures. I. Mendelsohn, Eric. 11. Series. 111. Series: North-Holland mathematics studies; 6 5 QA164.A4 511'.6 81-22594 ISBN 0-444-86365-6(Elsevier North-Holland) AACR2

PRINTED IN T H E NETHERLANDS

Dedicated to my parents Nathan Saul Mendelsohn Helen Brontman Mendelsohn

NATHAN SAUL MENDELSOHN

Nathan Mendelsohn was born on April 14, 1917 in Brooklyn, New York, and soon after moved to Canada (at age six months). Raised in Toronto, Nathan went on t o take B.A., M.A. and Ph.D. degrees from the University of Toronto finishing in 1942. One of the first things that anyone notices on a visit to the Mathematics Department at the University of Toronto is a plaque in the hall just outside the main departmental office honoring the winning team of the first William Lowell Putnam competition held in 1938. The University of Toronto was the winning team that year and the members of the winning team were Nathan Mendelsohn, Irving Kaplansky and John Coleman; which, when you stop t o think about it, is not a bad way to start out o n a career in mathematics! From 1942 through 1945 Nathan was engaged in research related to the war effort as a Research Scientist for the Defense Research Board of Canada (for example, as a member of the National Research Council Propellants Subcommittee of the Committee on Explosives). After the war Nathan taught for two years at Queen’s University before settling at the University of Manitoba. Nathan Mendelsohn’s contributions t o mathematics in Canada as a teacher, administrator, editor, international delegate, ‘server-on-committees’, elected officer, board member and moving force in general are well-known. (Perhaps less well-known, but equally important is Nathan’s ability as a magician. His up-close card tricks are of professional quality.) However, impressive as all of this is, his principal contribution is a large amount of important research in combinatorial mathematics, in recognition of which he was awarded in 1979 the Henry Marshall Tory Medal of the Royal Society of Canada. Probably the best measure of the significance of a mathematician’s research is the attention that it attracts. While it is not particularly difficult to write a lot of papers it is quite another matter to write even one paper which anyone ever reads. By any reasonable standard Nathan Mendelsohn has written a lot of papers which a lot of people have read. Over the past thirty-five years or so he has turned out a steady stream of papers (around a hundred) on Steiner systems and generalizations, orthogonal and perpendicular latin squares, all sorts of block designs, quasigroups, and all sorts of relationships between block designs, graphs and quasigroups. These papers have attracted attention and, in fact, have opened up new areas of research in combinatorial mathematics. For example, Nathan was the first to show that a decomposition of the complete directed graph 0,into edge disjoint triangles (all edges in the same direction) is possible if and only if u = 0 or 1 (mod 3), except u = 6. Today such decompositions are called Mendelsohn triple systems. In a subsequent paper Nathan constructed an infinite class of Mendelsohn triple systems which, if vii

viii

Nathan Saul Mendelsohn

order is disregarded, can be separated into a pair of perpendicular Steiner triple systems. Mendelsohn triple systems caught on and today at least fifty papers have been written on the subject. The history of Mendelsohn triple systems is typical of Nathan’s work, i.e., he has pioneered many avenues of research in combinatorics. I think it is safe to say there is not a combinatorist in the world who has not heard of Nathan Mendelsohn. I would also venture to say that, at least in the area of combinatonal algebra, there are very few who have not quoted at least one of his papers. Nathan is a remarkable mathematician in the two ways that really count: depth of scholarship and perception of the future. He is a remarkable human being in more ways than it is possible t o count. Happy birthday, Nathan, and many, many more. Curt Lindner Auburn University

NATHAN MENDELSOHN

As I HAVE KNOWN HIM

Writing these few pages about Nathan Mendelsohn takes me back to the days before the war. It was an exciting time with many celebrated people on staff and many students who have since made their names in the mathematical world. Brauer came to Toronto in 1935-in the same year Nathan entered the first year of the M&P course. Coxeter joined the staff in 1936 so that algebra and geometry were well taken care of. Analysis with Webber and applied mathematics with Synge and Infeld were going ‘full blast’. I have checked with the University Records office and found that Nathan earned First Class Honours in each of his undergraduate years, standing first on graduation in 1939 with a mark of 94. It was in 1938 that the William Lowell Putnam Competition was inaugurated and our team consisting of John Coleman, Irving Kaplansky and Nathan Mendelsohn stood first. The three members have distinguished themselves beyond most if not all our graduates (cf. the Mathematics Department at the U. of T. (U. of T. Press 1979)). Nathan’s career was complicated by the war which broke out in 1939. He began his graduate work with Brauer, but it was a year later than Brauer went on leave of absence to Michigan so it was I who became Nathan’s supervisor. Then I went to Ottawa to work with the National Research Council in the spring of 1941 so that Nathan’s Ph.D. thesis was largely his own effort; it was entitled “A group theoretic characterization of the general projective group” and accepted for the Ph.D. degree in 1942. Nathan had been interested in geometry as an undergraduate and I greatly enjoyed having him in my classes. This bridging the gap between algebra and geometry was to be characteristic of much of his later work. It is not necessary for me to list the more than 90 papers published since those early days, but it is interesting to watch his passage from one area to another. By 1958 he was involved in graph theory and latin squares but his love of matrix theory kept reappearing. Latterly he has worked largely in combinational mathematics. Nathan has had a great variety of responsibilities during his very productive life. H e worked with the National Research Council during the later years of the war 1942-45, he was elected a Fellow of the Royal Society of Canada in 1956, was Director of the Computing and Data Processing Society of Canada 1960-65, President of the Canadian Mathematical Society 1%9-71, Member of the Lowell Putnam Competition Committee 1970-72 and its Chairman in 1973 as well as being a member of the Department of Mathematics at the University of Manitoba and Chairman.

Nathan Mendelsohn. As I have known him

X

His latest honour was the Tory medal of 1979 presented by the Royal Society of Canada, and I would like to quote a paragraph from the letter nominating him for the award: Professor Herbert J. Ryser, a senior mathematician of the California Institute of Technology evaluates him in the enclosed letter “as one of Canada’s most outstanding mathematicians and, indeed, one of the most outstanding mathematicians in the world”. Professor Trevor Evans (Emory University), one of the leading experts in this field of research, writes: “It is a fantastic achievement to have accomplished so much first class work while carrying a major administrative burden for so long”. It is difficult to single out passages from the lavish praise heaped on Dr. Mendelsohn in the enclosed letter by Professor Charles C. Lindner (Auburn University), a prominent expert in the field. Here are a few samples: “Mendelsohn’s contribution t o mathematics in Canada as a teacher, administrator, ‘server-on-committees’, elected officer, board member and moving force in general is well known”. “Over the past fifteen years he has turned out a steady stream of incredibly innovative papers”. “These papers have attracted so much attention, so many mathematicians have become interested in the type of combinatorics in them that it is safe to say, they are the genesis of the branch of combinatorics known today as combinatorial universal algebra (or combinatorial algebra)”. “I think it is safe to say there is not a combinatorist or universal algebraist in the world who has not heard of Nathan Mendelsohn. I would also venture to say that there are probably very few (at least in combinatorics) who have not quoted at least one of his papers or worked in an area of research which he has helped to develop”. In nominating him Professor Aczel of Waterloo and I felt proud indeed and I consider it an honour to recall my long contact with him. He was married just after receiving his Ph.D. in Toronto and my wife and I had the pleasure of seeing the two of them in Ottawa o n their way to Quebec. He was awarded t h e Tory medal at the Royal Society Meeting in Saskatoon last June and this was a time of rejoicing for us all.

G. de B. Robinson*

N.S.Mendelsohn

was G . d e B. Robinson’s first Ph.D. student.

EDITOR’S PREFACE When I first decided to undertake this project, the problem of exactly which subjects would be considered arose. I sidestepped this question with the non-answer “the kind of mathematics that my father does and enjoys”. What a broad range that encompassed within combinatorics-latin squares, designs, groups of graphs, matchings, lattices, geometrics, etc.-all linked together by the interests of one man. What links these papers in combinatorics together is more than that. They reflect his approaches to combinatorics and in their totality yield a way of looking at combinatorics. It is at the interface between geometry (especially finite geometries) and universal algebra where this approach lies. I have not undertaken to write an introduction but have asked Professor G. de B. Robinson and C.C. Lindner to give their views. I thank them both very much for their time and thoughtfulness. I would like to thank also the contributors and referees for doing such a fine job. Each paper was most thoughtfully created and most diligently refereed. If there are any who ought to have been included in this volume and were not invited to contribute, I apologize. As consolation, I can only say that it is my hope that, G-d willing, I can solicit further papers at intervals for a sequel to this volume until my father’s one hundred-and-twentieth birthday. Finally, I would like to thank MIS. Martha Jarrell who did the typing and some of the organization and North-Holland Publishing Company for bearing with the delays. Above all, of course, my wife Lillian who put up with me during the not infrequent frustrations and gave me encouragement throughout this undertaking. Eric Mendelsohn

CONTENTS Nathan Saul Mendelsohn (C.C. LINDNER)

vii

Nathan Mendelsohn. As I have known him (G. DE B. ROBINSON)

ix

Editor's preface

xi

B. ALSPACH and T.D. PARSONS, On hamiltonian cycles in metacirculant graphs

1

L.D. ANDERSEN, Embedding latin squares with prescribed diagonal

9

L.D. ANDERSEN and E. MENDELSOHN, A direct construction for latin squares without proper subsquares

27

L. BABAIand J. NESETRIL,High chromatic rigid graphs I1

55

F.E. BENNET,Direct constructions for perfect 3-cyclic designs

63

N.L. BIGGS,Distance-regular graphs with diameter three

69

P.J. CAMERON, Colour schemes

81

M.J.COLBOURN and C.J. COLBOURN, The analysis of directed triple systems by refinement

J. DBNESand P. HERMANN, On the product of all elements in a finite DOUP

97 105

R.H.F. DENNISTON, Enumeration of symmetric designs (25,9,3)

111

P. E M S and J. LARSON, On painvise balanced block designs with the sizes of blocks as uniform as possible

129

T. EVANS, Finite representations of two-variable identities or why are finite fields important in combinatorics ?

135

T. EVANS and M. FRANCEL, Some connections between Steiner systems and self-conjugate sets of m.o.1.s.

143

U. FAIGLE, Incidence-geometric aspects of finite abelian groups

161

D. GALEand A.J. HOFFMAN, Two remarks on the Mendelsohn-Dulmage theorem

171

B. GANTERand R.

179

QUACKENBUSH,

Duroids xii

Contents

...

XI11

H. GROFLIN and A.J. HOFFMAN, Lattice polyhedra 11: Generalization, construction and examples

189

W. HAEMERS and J.H. VAN LINT,A partial geometry pg(9, 8, 4)

205

Z, HEDRL~N, P. HELL and C.S. KO, Homomorphism interpolation and approximation

213

K. HEINRICH,Prolongation in rn -dimensional permutation cubes

229

A.J.W. HILTON and C.A. RODGER,Match-tables

239

and A.D. KEEDWELL,On the sequenceability of dihedral G.B. HOGHTON groups

253

A combinatorial construction of the small Mathieu D.R. HUGHES, designs and groups

259

C.C. LINDNER and W.D. WALLIS,Embeddings and prescribed intersections of transitive triple systems

265

R. MATHON,On linked arrays of pairs

273

E. MENDEUOHN and K.T. PHELPs, Simple Steiner quadruple systems

293

A. NEUMAIER, Rectagraphs, diagrams, and Suzuki’s sporadic simple

group

305

R. PADMANABHAN, Logic of equality in geometry

319

F. PIPER,On axial automorphisms of symmetric designs

333

I. &VAL and B. SANDS, Pictures in lattice theory

341

S.A. VANSTONE, On mutually orthogonal resolutions and nearresolutions

357

Appendix. Research papers by N.S. Mendelsohn

371

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Annals of Discrete Mathematics 15 (1982) 1-7 @ North-Holland Publishing Company

ON HAMILTONIAN CYCLES IN METACIRCULANT GRAPHS Brian ALSPACH* and T.D. PARSONS** Dedicated to Nathan Mendelsohn on the occasion of his 65th birthday In this paper it is shown that every connected metacirculant with an odd number of vertices greater than one and with blocks of prime cardinality has a hamiltonian cycle.

1. Introduction

There are only four known nontrivial vertex-transitive graphs which fail t o have a harniltonian cycle. L. Babai [3] has asked whether there are infinitely many such graphs; C. Thomassen [4, p. 1631 has conjectured that there are not. The four examples are the Petersen graph, the Coxeter graph [5, p. 2411 and the graphs obtained from these two by replacing every vertex by a triangle. None of these four graphs is a Cayley graph, so that it might be conjectured that every connected Cayley graph for a finite group K has a hamiltonian cycle-and this has been shown true at least for abelian groups K and some other special groups [6,8]. Metacirculant graphs were introduced in [2] as an interesting class of vertex-transitive graphs which included many non-Cayley graphs and which might contain further examples of non-hamiltonian graphs. Their construction generalized a construction in [l] and [7] which included the Petersen graph and some other non-Cayley graphs. It is reasonable to ask whether or not every connected metacirculant except the Petersen graph contains a hamiltonian cycle. This question is considered here for those metacirculants consisting of an odd number of ‘blocks’ of prime order. A theorem of Chen and Quimpo, of great significance t o the general problem of hamiltonian cycles in connected vertex-transitive graphs, is applied to obtain hamiltonian cycles in the rnetacirculants under consideration. It is

‘This research was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-4792. * * This research was partially supported by the National Science Foundation under Grant MCS-8002263. 1

B. Akpach, T.D. Parsons

2

hoped that the techniques used here may be useful for the general problem itself. Before proceeding t o the main result, a brief review of metacirculants is in order. Let Z,, denote the ring of integers modulo n and Z : denote the multiplicative group of units in Z,,. Choose positive integers m and n, let LY E Z : and let p always denote the integer part of m/2. If A C Z,,, then -A = {-u: a E A } and aA = { a a : a E A}. The (m, n)-metacirculant graph G ( m , n, a, So, S,. . . . , S,) has vertex-set ( u j : 0 < i m - 1 and 0 S j n - 1) and edges defined by the sets Si, 0 S i S p, provided that the following hold: 0 fZ S,>= -So

and

S i C Z , , and

and

SoCZ,,

(1)

amSI= Si for O S i S p

if m is even, then a'S,

= -S,

.

The edges are then defined by u: is adjacent to u:

if and only if

s - r E a'si+

(4)

where we assume 0 =sj - i S p and all differences are computed in 2, or 2, as appropriate. There are several observations t o make about the metacirculant G = G ( m , n, a,So, S,, . . . , S&). First, the construction is designed to allow the permutations p = (ugup. . ' u".,)(ubu;

..

*

Ui-,)

. ..

..

*

u;--/)

(5)

and r, defined by T(U:) = U $ ',

(6)

to be automorphisms of G. Thus, G is vertex-transitive. Second, once we know what vertices are adjacent t o u8 (as described by SO,S1, . . . , S,), we know all the edges in the graph from the automorphisms p and 7 (as described in (4)). 'Third. this construction produces a variety of interesting vertex-transitive graphs. For example. G(2,5,2, {1,4}, (0)) is the Petersen graph and, as mentioned earlier, infinitely many non-Cayley graphs are produced.

2. Main result

The principal purpose of this paper is t o prove the following theorem about hamiltonian cycles in (m. n)-metacirculants when rn is odd and n is prime. We now state the main result.

On hamiltonian cycles in metacirculant graphs

3

Theorem 1. If G = G(m, n, a,So, S1, . . . , S,,)is a connected rnetacirculant graph with rn odd, n a prime and at least 3 vertices, then G has a hamiltonian cycle. The proof of Theorem 1 requires several results whose statements we include for the sake of completeness. One should recall that the Cayley graph G ( K ,H ) is the graph whose vertex-set consists of the elements of the group K with g adjacent to g' if and only if g' = gh for some element h in H where H K and 1 f2H = H-'. A graph G is said to be hamiltonian connected if for any two vertices u and u of G, there is a hamiltonian path whose terminal vertices are u and u. The next result is frequently used, and was first proved by Chen and Quimpo [6].

Theorem 2. If G is a connected Cayley graph of an abelian group of order at least 3, then G is hamiltonian connected if and only if G is regular of degree at teast 3 and G is not bipartite. In the case that G is bipartite and is regular of degree at least 3, there is a hamiltonian path joining any two vertices in different bipartition sets.

If G is a connected Cayley graph of an abelian group of order at least 3, then every edge of G lies in a hamiltonian cycle.

Corollary 3.

A semidirect product of a group K by a group L is a group M such that K is a normal subgroup of M, L is a subgroup of M, K and L have only the identity element in common, and K U L generates all of M. The next result has been proved by MaruSiE [S].

Theorem 4. If G is a connected Cayley graph of a semidirect product of a cyclic group of prime order by an odd order abelian group, then G has a harniltonian cycle. The final result that we mention in this section is the following sufficiency condition for a metacirculant graph to be a Cayley graph. It is proved in [2].

Theorem 5. Let G = G(m, n, a,So,S1,.. . , S,,),a be the order of a in Z : and c = a/gcd(a, rn). If gcd(c, m ) = 1, then G is a Cayley graph for the group (p, 7 ' ) . Furthermore, this group is abelian if gcd(a, m ) = 1. We mention here that the group (p, +) is a semidirect product of the cyclic group ( p ) by the cyclic group ( 7 ' ) . This fact will be useful in the following material.

B. Afspuch, T.D. Parsons

4

3. Proof of the main theorem We now embark on t h e proof o f the main theorem. We shall always assume that m is odd. n is prime and G is connected, with at least 3 vertices. We shall be performing all computation in either 2, or 2, and it will be clear from the context which is which. The proof consists of restricting the class of possible non-hamiltonian metacirculants t o a smaller and smaller class until everything has been shown to have a hamiltonian cycle.

b m m a 6. If m

= 1,

then G has a hamiltonian cycle.

Proof. If m = 1, then G is in fact a circulant graph, that is, a Cayley graph o n t h e cyclic group C,, so t h e result follows from Corollary 3. Lemma 7. lf m = 3, then G has a hamiltonian cycle.

Proof. This is proved in [Y] Lemma 8. If n = 2. then G has a hamiltonian cycle. Proof. Since n = 2 and a E Z t , the only possibility for a is a = 1. Then G is a Cayfey graph for an abelian group by Theorem 5, and the result follows from Corollary 3. Lemma 9. If a m= 1 or a

=

-1, then G has a hamiltonian cycle.

Proof. If a m= 1, then the integer c in Theorem 5 has the value c = 1. If instead a = -1. then c = 2. In either case, by Theorem 5 G is a Cayley graph for the group (p, 7 ' ) . As mentioned at the end of Section 2, this group is a semidirect product of a prime order cyclic group by an odd order abelian (cyclic) group. It follows from Theorem 4 that G has a hamiltonian cycle. Because of Lemmas 6. 7 and 8 we assume in the remainder of the proof that n is an odd prime and rn is at least 5. Some further nomenclature is also required. For i = 0, 1, . . . , rn - 1, let U' = {ub, U;,. . . , u:-'} and let GI = G [U ' ] denote the subgraph of G induced by U'. Note that GI is the circulant graph with symbol a'so. Let G / p denote the graph with vertex-set { U',U ' , . . . , Urn-'} and U' adjacent to U' if and only if Sl-, f 0 where we assume 1G j - i S p. In other words. G I p i s the circulant graph with rn vertices and symbol S = {*i: 1 G i S p and Sif 0).

On hamiltonian cycles in metacirculant graphs

5

Lemma 10. Glp is connected and every edge of Glp lies in a hamiltonian cycle in Glp.

Proof. It is clear that Glp being connected is implied by the fact that G is connected. The result now follows from Corollary 3 because Glp is a connected circulant graph on m > 3 vertices.

Lemma 11. If Gois regular of degree at least 3, then G has a hamiltonian cycle.

- -

Proof. Let VU'1 * W-lV be a hamiltonian cycle in Glp by using .Lemma . 10. Since n 2 3 and p E Aut(G), there exist edges U ~ U ; , ,u j ~ u f z.,. . , u ' ,,m-2 m - 2 u ~ mkm+ -l "-I ,n-lu& in G no two of which have a vertex in common. Now, Gois a circulant graph of odd prime order n and degree at least 3, therefore Go is connected. By Corollary 3, then Gohas a cycle of (odd) length n, so Go is not bipartiteand is thus hamiltonian connected. Thus, each Gi, i = 0,1, . . . , m - 1, is hamiltonian connected as well. Now let 8; be a hamiltonian path in Gi,with terminal vertices ui, and uff where io = 0 = jo. Then ( ~ ! u ~ ~ ) p i , ( u *~- ~* (ui'"-Iuo u I~m -~i )kiP)Po , is a hamiltonian cycle in G.

Because of Lemma 11 the remainder of the proof is concerned only with the cases that S,,= 0 or [&I= 2. The next result also places severe restrictions on the cardinalities of the Si'sfor i = 1 , 2 , . . . , p.

Lemma 12. If ISi/3 2 for any i satisfying 1=si =zp, then G has a hamiltonian cycle.

-

Praof. Let LJ"U'1Ub- * U i m - l u be a hamiltonian cycle in GIp such that i,,-l = i where ISi/3 2. The existence of such a hamiltonian cycle is guaranteed by Lemma 10. Now let uSuf:u; - - u:::ui,-, be a corresponding path in G where uk is a vertex of the set Uirfor r = 1,2, . . . , m - 1. Since ISi/2 2 there is U : . n an edge from u;,-~ to uy where u: # u!. Let P be the path ugui: . * * U ~ ~ - ~ Since is a prime, P upr(P)u ~ * ' ( PU ). - UP("-')'(P) is a hamiltonian cycle in G.

Lemma 13. If some Si satisjies s E Si with s # 0, then G has a hamiltonian cycle if l s i a p .

Proof. By Lemma 12 we may assume that ISi(= 1 and by (2) we know amSi= Si. From this we conclude that am= 1 and the result follows from (10). By Lemma 13 we know that each S,, 1 S i sz p, is either empty or Si = (0). But this implies that So # 0 because G is connected. By Lemma 11 we know

B. Alspch, T.D. Parsons

6

that /Sol= 2 so that Go is itself an n-cycle. Since amso= SOmust hold by (2), either a m= 1 or a m= -1. If a m= 1 or a = -1, we are done by Lemma 9. Thus we may assume that a m= - 1 and a # -1. which implies that

Now let So = {k, - k } and VV'l . . Uim-lU'be a hamiltonian cycle in Glp. Then consider the paths P6 for i = 0, 1 , . . . , m - 1 in G; given by

Let PI denote the path pf(Pb) so that its initial vertex is uf and its terminal vertex is either u;_& or u;+,ak. Now consider the path P in G given by P = P ~ ( u ~ u ~ ) P ~ ( u ~. .~P::: u ; ~with ) P ;initial ~ . vertex u8 and terminal vertex ulr::. Here

for every

t=

I , . . . , m - 1. In particular,

so that P together with the edge ub"-'ug forms a hamiltonian cycle in G. This completes the proof of the main theorem.

4. Additional comments

It should be noted that some of the results proved above hold for more general situations. For example, Lemma 11 is true for any odd n whenever Go is connected and is regular of degree at least 3 via essentially the same proof. Also, it is natural to ask whether or not the main theorem holds when rn is even, II is prime and G is not the Petersen graph. Most of the work for this case h a s been completed except for the case in which Is01 S 2 , ISi(=z 1 for i = 1 , 2 , . . . , p and G is a Cayley graph. Then, Theorem 4 cannot be employed since an even order abelian group is required. The most general result that seems plausible is that every connected

On hamiltonian cycles in metacirculant graphs

7

rnetacirculant with at least three vertices has a harniltonian cycle with the sole exception of the Petersen graph. Such a result would be worthwhile. Department of Mathematics Simon Fraser University Burnaby, Canada Department of Mathematics The Pennsylvania State University Pennsylvania, USA

References [l] B. Alspach and R.J. Sutcliffe, Vertex-transitive graphs of order 2p, Ann. N.Y. Acad. Sci. 319 (1979)19-27. [2] B. Alspach and T.D. Parsons, A construction for vertex-transitive graphs, Canad. J. Math., to appear. (31 L. Babai, Problem 17, in: Unsolved Problems, Summer Research Workshop in Algebraic Combinatorics, Simon Fraser University, 1979. [4] L.W. Beineke and R.J. Wilson, eds.,Selected Topics in Graph Theory (Academic Press, London, 1978). (51 J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Elsevier, New York, 1976). [6] C.C.Chen and N.F. Quimpo, On strongly hamiltonian abelian group graphs, Lecture Notes in Mathematics 884 (Springer, Berlin) to appear. [7]D. M a d i t , On vertex symmetric digraphs, Discrete Math. 36 (1981)69-81. [8] D.Marugit, Hamiltonian circuits in Cayley graphs, submitted. [9] D. Marugit, Hamiltonian circuits in vertex-symmetric graphs of order 3p, preprint.

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Annals of Discrete Mathematics 15 (1982) 9-26 @ North-Holland Publishing Company

EMBEDDING LATIN SQUARES WITH PRESCRIBED DIAGONAL Lars DBvling ANDERSEN Dedicated to Professor N.S. Mendelsohn on his 65rh birthday The paper is concerned with embedding incomplete latin squares in latin squares with prescribed diagonal. A new result about symmetric latin squares is proved and is then applied to improve a theorem of Hilton on idempotent latin squares and answer a question of Lindner about half-idempotent latin squares.

1. Introduction A latin square of side n is an n x n matrix, each of whose cells contains an element from a set of n symbols in such a way that each symbol occurs exactly once in every row and exactly once in every column. An incomplere latin square of side r on n symbols is an r x r matrix, each of whose cells contains an element from a set of n symbols in such a way that each symbol occurs at most once in every row and at most once in every column. A partial latin square of side r on n symbols is an r x r matrix, each of whose cells is either empty or contains an element from a set of n symbols in such a way that each symbol occurs at most once in every row and at most once in every column. In some cases, for example when regarding latin squares as quasigroups, it is natural to consider partial latin squares of side r t o be on r symbols. We shall not employ the terminology of quasigroups in this paper (the reader interested in this aspect is referred to [13]). We should note that the distinction between incomplete and partial latin squares explained above is not quite standard (sometimes the two words are used as synonyms). It is, however, convenient for our purpose. If L is a latin square of side n and 1 s r S n, then any r x r submatrix R is an incomplete latin square. We say that R is embedded in L. Similarly for partial latin squares. We shall usually think of R as being situated in the top left-hand corner of L (Fig. 1 gives the idea). This paper is concerned with embeddings of incomplete and partial latin squares R in latin squares, where (all or part of) the diagonal outside R is given (prescribed). The diagonal of a matrix is the sequence of symbols in cells 9

10

L.D. Andersen

Fig. 1

. . . . Thus, writing L = (l,]), the following problem is typical: given f, for all (i, j ) such that 1 S i ==r and 1s j S r and given 1, for r + 1 s 1s m for some m S n, can the remaining li, for 1 S i S n and 1S j S n be defined so as to (1, I),(2, Z),

obtain a latin square ? (See Fig. 1.) In particular, we shall be interested in idemporent latin squares, i.e., latin squares L for which all I,, are different. We also speak of idempotent incomplete and partial latin squares (in the latter case we require that all diagonal cells are non-empty and, of course, that the diagonal consists of distinct symbols). If L is an idempotent latin square on the symbols (1, . . . , n} we may think of the diagonal as being (1,. . . , n). We finally define a (possibly incomplete or partial) latin square L as being symmetric if = f], for all i and j for which any of the two is defined. We briefly survey some well-known results and contribute some new ones. The known results are stated in Section 2. In Section 3 we prove our main tool, a theorem about embeddings of incomplete symmetric latin squares (or rather incomplete externally symmetric latin squares, defined in that section) in latin squares with prescribed diagonal. In Section 4 we consider a theorem of Hilton about embedding idempotent partial latin squares, and improve it a little. Lastly, in Section 5 we answer a question of Lindner about embeddings of so-called half-idempotent latin squares and apply t h e result to Steiner Triple Systems.

2. Previous results

There are some classic results about embeddings of incomplete latin squares (or rectangIes, an incomplete larin rectungfe of size r X s on n symbols being an r x s matrix in which each cell contains an element from a set of n symbols in such a way that each symbol occurs at most once in each row and at most once in each column). Considering latin squares on symbols ( 1 , . . . , n} and defining N R ( j )to be the number of times that the symbol j occurs in the incomplete

Embedding latin squares with prescribed diagonal

11

latin rectangle R, it is common to these results that what matters are the numbers N R ( j ) and, in the case of symmetric squares, the symbols in the diagonal. We list the most important results below.

Tbeorem 1 [14]. A n incomplete latin rectangle R of size r x s on symbols (1, . . . ,n} can be embedded in a latin square of side n if and only if NR(j)>r+s-n

forallj, l s j s n .

Theorem 2 [5]. A n incomplete symmetric latin square R of side r on symboki (1, . . . , n } can be embedded in a symmetric latin square of side n if and only if N ~ ( j ) s 2 r n-

forallj, l s j s n

NR( j ) = n (mod 2)

for at least r symbols j .

and

From these two theorems, giving necessary and sufficient conditions for embedding incomplete latin squares, it is easy to deduce the following results stating sufficient conditions for the embedding of partial latin squares.

Theorem 3 [7]. For any r and any n 2 2r, any partial latin square of side r on n symbols can be embedded in a latin square of side n. Theorem 4 [5]. For any r and any euen n 3 2r, any partial symmetric latin square of side r on n symboki can be embedded in a symmetric latin square of side n. For any r and any odd n > 2r, a partial symmetric latin square of side r on n symbols can be embedded in a symmetric latin square of side n if and only if no symbol occurs twice in the diagonal of R. A symmetric latin square of odd side must have each symbol occurring exactly once in the diagonal (as each symbol occurs an even number of times in the non-diagonal cells of any symmetric square); hence Theorem 4 has the following corollary. Corollary 5 [5]. For any r and any odd n > 21, any idempotent partial symmetric latin square of side r on n symbols can be embedded in an idempotent symmetric latin square of side n.

The inequalities in Theorems 3 and 4 and Corollary 5 are best possible. Most of the results in Theorems 1-4 are consequences of Theorems 8 and 11 of this paper. They are generalized in another direction in [2].

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L.D. Andersen

When we turn to latin squares with prescribed diagonal, we do not get as nice results as Theorems 1 and 2. When given an incomplete or partial latin square R on symbols (1,. . . , n } and a non-negative integral-valued function f defined on { 1, . . . , n } we say that R is embedded in a latin square L of side n with prescribed diagonalf, if R is embedded in L and each symbol j occurs at least f ( j ) times in the diagonal of L outside R. If C,.l f(j) = n - r, the number of diagonal cells of L outside R, we say that the diagonal is completely prescribed. In the definitions above we have not required that the symbols occur in any prespecified order in the diagonal cells of L; this can, however, easily be obtained by permuting rows and permuting columns, so it is quite justified to speak of embedding with prescribed diagonal. It seems very hard to obtain necessary and sufficient conditions for embedding an incomplete latin square R of side r on symbols (1,. . . , n} in a latin square L of side n with completely prescribed diagonal f , and nobody has yet been able to do so. It is easy to see that the following condition is necessary (see t h e proof of Theorem 11, and Fig. 3).

Unfortunately, Condition 6 is not sufficient. Fig. 2 shows a simple example illustrating this. The incomplete latin square R of side 3 cannot be embedded in a latin square of side 5 with prescribed diagonal f , where f(1) = f ( 2 ) = f(3) = 0 and f ( 4 ) = f ( S ) = 1 .

Fig. 2.

It can be seen that embeddability with prescribed diagonal does not depend on the numbers N R ( j )alone. There is a discussion of this in [l], which also contains the following theorem. Theorem 7. Let n 3 IS. For each r, Ln/2J < r < n - 1, there exists an idempotent incomplete latin square R of side r on symbols (1,. . . , n } satisfying Condition 6 which cannot be embedded in an idempotent latin square of side n.

Embedding latin squares with prescribed diagonal

13

In [ 11 the incomplete latin squares whose existence is ensured by Theorem 7 are actually constructed, and similar results are stated for other prescribed diagonals, not necessarily corresponding to idempotent squares. Ref. [l] also contains the following result, which later appeared in [3], and which states that Condition 6 is necessary and sufficient for the required embedding, if only the diagonal is not completely prescribed. Thus, if at least one place in the diagonal is left unprescribed, we know exactly when embedding is possible.

Theorem 8 [3]. A n incomplete latin square R of side r on symbols (1, . . . , n} can be embedded in a latin square of side n with prescribed diagonal f , where X ~ = l f ( j ) < n - r - l ifandonlyif , & ( j ) 2 21 - n + f ( j ) for all j , 1s j s n

This leaves the problem of characterizing those incomplete latin squares which are embeddable in latin squares with completely prescribed diagonal (in the next section we solve this problem for symmehic latin squares). Most attention has been given to the case where the incomplete latin square is to be embedded in an idempotent latin square. And in this case there is another major unsolved problem, namely to obtain a best possible result corresponding to Theorems 3 and 4. Hilton made the following conjecture. Conjecture [8]. For any r and any n 2 21 + 1, any idempotent partial latin square of side r on r symbols can be embedded in an idempotent latin square of side n. We note that since an idempotent symmetric latin square must be of odd side, Corollary 5 states the truth of the conjecture for symmetric latin squares. It follows from Condition 6 that the inequality n 2 2 r + 1 would be best possible, but it is interesting to observe that, by Theorem 7, it would also be best possible for ‘other reasons’ than the obvious Condition 6. The author believes that Hilton’s conjecture is true, also if the idempotent partial latin square of side r is allowed to be on n symbols. It would suffice to prove this for idempotent incomplete latin squares, because suppose that R is an idempotent partial latin square of side r on n symbols, where n 3 2r + 1, and suppose that R is not incomplete. Then we can fill any empty cell of R with one of the n symbols, because there will be at least n - 2 ( r - 1 ) 2 3 symbols available. By repeated use of this argument we can get an idempotent incomplete latin square. The conjecture is still open. Lindner [ll] proved that any idempotent partial

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L.D. Andersen

latin square can be embedded in some idernpotent latin square, and Hilton has proved the following result. which is the best result so far in this direction. Theorem 9 [S]. For any r and any k 3 0 , any idempotent partial latin square of side r on r symbols can be embedded in an idempotent latin square of side ti = 4r + 4k.

As a matter of fact, Hilton’s proof works if the partial square is on n/2 symbols. Hilton noted that Theorem 9 implies that an idempotent partial latin square of side r on r symbols can be embedded in an idempotent latin square of side n for any n 3 8r. We explain this in Section 4, where we also discuss Theorem 9 in more detail-we improve it to all n z= 4r (except n = 4r + 1). W e finally remark that Theorem 8 has the following corollary, showing that in a certain sense Hilton’s conjecture is ‘almost true’. Corollary 10. For any r and any n 2 2r + 1, any idempotent partial lafin square of side r on n symbols can be embedded in a latin square of side n which has at least n - 1 distinct symbols in the diagonal.

3. Embedding incomplete externally symmetric latin squares with prescribed diagonal

Whereas n o generalization of Theorem 1 to embeddings with prescribed diagonal is known (Theorem 8 is the best we can do, but it does not work for completely prescribed diagonal), we now prove a theorem which is a generalization of Theorem 2, also covering the case with completely prescribed diagonal. In order to be able to embed a little more than just incomplete symmetric latin squares, we define an incomplete latin square of side r t o be externally symmetric if the same symbols occur in row i as in column i for all i, 1 6 i S r. Thus a symmetric square is also externally symmetric, but the incomplete latin square of side 3 in Fig. 2 is externally symmetric and not symmetric. If an incomplete latin square R is embedded in a latin square L we say that L is symmem‘c off R if (R is in the top left-hand corner of L and) fij = lji for all cells ( i , j ) and ( j , i ) not in R. It follows, that R would have to be externally symmetric. Theorem 11. A n incomplete externally symmetric latin square R of side r on symbols ( 1 , . . . , n } can be embedded in a latin square of side n, which is

Embedding latin squares wirh prescribed diagonal

15

symmetric off R, with prescribed diagonal f, if and only if

and

N R ( j )3 21 - n + f(j) for all j , 1 ~j s n ,

N R ( ~+ )f(j)= n (mod 2) for at leust r +

I= f(i) symbols j . n

i= 1

Proof. Necessity of the conditions. Assume that R is embedded in a latin square L as required. Let L be partitioned as indicated in Fig. 3. Then we have, for each symbol j ,

This proves the necessity of the first condition. r

n- r

Fig. 3.

For each j , 1Sj =S n, let g ( j ) be the number of times that the symbol j occurs in the diagonal of B. Since each symbol occurs an even number of times in the non-diagonal cells of L outside R, and n times in L altogether, it follows that

+

N R ( j ) g ( j ) = n (mod 2 )

for all j, 1 =S

n.

But g(j)#f(j) for at most n - r-XYZ1f(i) symbols j , so there are at least n - ( n - r - XY=l f(i)) symbols j for which

which proves the necessity of the second condition. Sufliciency of the conditions. In this part of the proof we shall use a little bit of graph theoretical terminology and one result from the theory of edgecolourings of graphs. The terminology is standard; it may be found in [4].

L.D. Andersen

16

Let R be an incomplete externally symmetric latin square of side r on symbols (1. . . . n} and let f be given, satisfying the conditions of the theorem. We may assume that C;=,f ( j ) = n - r (i.e., that the diagonal is completely prescribed). Because if Z;=, f ( j ) < n - r put f ’ ( j ) = f ( j ) + 1 for all j for which N R ( j ) + f ( j ) f n (mod2),f’(j)=f(j)otherwise;thenC.i”=lf(j)sC,”=lf(j)+ (n ( r + Z y - z lf(i)))= n - r a n d N R ( j ) z 2 r - n+f(j)andNR(j)+f’(j)=n(mod2)for all j. 1 s j n : furthermore, n - r - Zyrlf’(j)= n - r-Z,”=l ( N R ( j ) -n)= n - r rz + n’ = 0 (mod 2) and c;=,N R ( j )- Z;=, (2r - n + f ’ ( j ) ) = r’ - 2rn + n2 C ; = , f ( j )= ( n - r ) 2 - C ; = , f ’ ( j ) > n - r-C;,,f’(j),soitfoIlowsthatwecandefine f” such that f ’ ( j ) = f ” ( j ) (mod 2) and N R ( j )2 2r - n + f”(j) for all j , I S j S n, and E;-.l f ” ( j ) = n - r. Then we could work with p’ instead of f. Henceforth we shall f(j) = n - r. We also assume that r < n. a s u m e that C,”=, We shall extend R to an incomplete externally symmetric latin square R I of side r + 1 and produce a function f i such that we get the required embedding by embedding R 1in a latin square of side n, symmetric off R1, with prescribed diagonal f l . and such that R 1and f l satisfy the conditions of the theorem (with r + 1 in place of r). Repeated use of the argument then gives the desired embedding. } {m, . . . , UJ, We form a bipartite graph G with vertex set {pl, . . . , pn P , + ~U where, for 1 i r and 1 s j n, pi is joined to ujby a single edge if and only if the symbol j does not occur in row i of R, and where P , + ~is joined by f ( j ) edges to each q. Then the degree d ( p i ) = n - r for all z, 1 S i S r + I, and d ( m j ) = r - N R ( j ) + f ( j ) s r - ( 2 r - n + f ( j ) ) + f ( j ) = n - r for each j, 1 S j s n (see Fig. 4). By a theorem of Konig [lo], G has a proper (n - r)-edge-colouring, i.e., a colouring of t h e edges such that no two edges incident with the same vertex have the same colour, and such that n - r distinct colours are used. Consider

.

ssinR

degrees n-r

Fig. 4.

Embedding latin squares wifh prescribed diagonal

17

such an, edge-colouring, and let k be a symbol for which f ( k )> 0 (one exists, as r < n). Let c be the colour of some edge joining ukto P , + ~ . Now place symbol j in cell (i, r + 1) of R1 if and only if ujis joined to pi by an edge of colour c, for all i and j . Then one symbol is placed in each cell of column r + 1, because each colour, and hence also the colour c, occurs on exactly one edge incident with each vertex pi, 1 s i S r + 1. Notice that the symbol k is placed in cell ( r + 1, r + 1). Also put the symbol placed in cell (i, r + 1 ) into cell ( r + 1 , i ) for each i, 1 S i 6 r (the symbol is missing from column i of R, because R is externally symmetric). Clearly R1 obtained in this way is an incomplete externally symmetric latin square and the entries not in R are symmetrically placed. Put fi(k) = f ( k ) - 1 , and f l ( j ) = f ( j ) for all j # k. Clearly N R , ( ~ + )f l ( j ) = n (mod 2) for all symbols j. (And Z,”=lfl(j)= n - r - 1 = n - (r + l).) We only have to show that each symbol j occurs at least 2(r + 1)- n + f l ( j ) times in R1. This is true for the symbol k, because

NR,(k) = N R ( k ) + 1a (2r - n + f ( k ) ) + 1 = 2r - n + ( f l ( k )+ 1) f 1 =2(r+1)-n+fl(k). Let j # k . Then 2 ( r + 1 ) - n + f l ( j ) = 2 r + 2 - n + f ( j ) . If j occurred 2 r + 2 n + f ( j ) times in R, then clearly it occurs at least that many times in R l . So assume finally that N R ( j ) < z r + Z - n + f ( j ) . s i n c e & ( j ) = f ( j ) - n (mod2)and N R ( j )3 2r - n + f ( j ) it follows that N R ( j ) = 2r - n + f(j).Then the degree of aj in G is exactly n - r, and so the colour c occurs on an edge incident with aj; since j # k, it is not an edge joining q to P , + ~ Hence . the symbol j occurs once in column r + 1 of R1, and once in row r + 1, and we have that N R , ( j )= NR(j ) + 2 = 2r + 2 - n + f(j), as required. This concludes the proof of Theorem 11.

Corollary 12. Let R be an incomplete symmetric latin square of side r on symbols (1, . . . , n}, and let h ( j ) be the number of times that j occurs in the diagonal of R, 1 =sj s n. Then R can be embedded in a symmetric latin square of side n with prescribed diagonal f if and only if and

N R ( j )3 2r - n + f(j)

for all j , 1 S j S n ,

h ( j ) + f ( j ) = n (mod 2) for at least r

+ XG1f ( i ) symbols j .

The result of Corollary 12 (with completely prescribed diagonal) was first proved by Hoffman [9], using exactly the same technique as the one used in the proof of our Theorem 11.

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L.D. Andersen

4. Application to Hilton’s result

As stated in Theorem 9, Hilton proved that an idempotent partial latin square of side r on 4 2 symbols can be embedded in an idempotent latin square of side n = 4r + 4k, for any k 3 0 . In the same paper he proved the following theorem.

Theorem 13 [8]. For any n and any t # n, an idempotent latin square of side n can be embedded in an idempotent latin square of side t if and only if t 2 2n + 1. Putting k = 0 in Theorem 9 and then applying Theorem 13 we get the following corollary. Corollary 14 [8]. For any r and any n 2 81, any idempotent partial latin square

of side r on 21 symbols can be embedded in an idempotent latin square of side n. Hilton’s conjecture states that the inequality n 2 8 1 can be improved to n 2 2 r + 1. We shall improve it to n 2 4 r + 2 . Thus the embedding will be proved possible for all n 2 4r except n = 41 + 1; the improvement on Hilton’s results presented here consists of all n for which 4 r + 2 S n < 8 r and n f O (mod 4). The proof of the following theorem falls into two cases: n odd and n even. The proof for n odd applies Theorem 11, whereas the proof for n even is just a slight alteration of Hilton’s proof of Theorem 9.

Theorem 15. For any r and any n 3 41, n # 41 + 1, any idempotent partial latin square of side r on [n/2J- 1 symbols can be embedded in an idempotent latin square of side n. Proof. If r = 1 the statement of the theorem becomes merely the existence of an idempotent latin square of side n, which is well known for all n 3 3. S o we assume that r 3 2. n odd. Let t = 2 [ ( n+ 1)/4J - 1. Then t is odd, t 2 ( n - 1)/2- 1 and n is 2t + 1 or 2t + 3. Let R be an idempotent partial latin square of side r on t symbols. As r 2 2 L((4r + 3) + 1)/4] - 1 = 21 + 1, Corollary 10 can be applied, and we can embed R in a latin square T of side t in which t - 1 distinct symbols occur in the diagonal. If all t symbols occur in the diagonal of T, it is easy to see that the conditions of Theorem 11 are fulfilled, and so T can be embedded in an idempotent latin square of side n (of course, this embedding also follows directly from Theorem 13). So we suppose that only t - 1 distinct symbols occur in the diagonal of T.

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Embedding latin squares with prescribed diagonal

We can assume that the diagonal of T is (1,2,. . . , t - 1, a), where a E {I,. , .,t - 1); hence the symbol f does not occur in the diagonal of T. Now we change the entry of cell (t, t) of T to obtain T' and extend it to an idempotent incomplete latin square S of side t + 3 on symbols (1, . . . , n} as shown in Fig. 5 (since r 3 2 we have f 3 5 and n B t + 6). In both of the different cases, D is a 3 X r array.

I

-$

T' Sr

n t+4 t + 5 t+l t + 2

t + 6 t+

D if n = 2 t + ?

3 DT

t+5 t+6

t

t+7

t+2

... ... ... D

n-2 n-q

n t+2 a

n-1 n t + 4 t + 3 n t+4 t+

if n = 2 t + 3

Fig. 5.

It is easy t o see that S is indeed an incomplete latin square and that in fact it is externally symmehic. We also have N S ( a ) =t t - 2 , N s ( j )= t , for all j E (1, . . . , t - l}\{cr}, N,(t)= t + 3 ,

5 ifn=2t+1, N s ( j )= 3 i f n = 2 t + 3 .

[

Ns(t + 4) =

and

[75

for all j E {t + 1, t + 2, t + 3},

ifn=2t+1, ifn=2f+3,

6 ifn=2t+1, for all j E {t + 5, t + 6) N s ( j )= 4 i f n = 2 t + 3 ,

[

N s ( j )= 6 for all j E { t + 7 , . . . , n} .

We only have left to observe that we can apply Theorem 11 to embed S in an idempotent latin square of side n. We put f( j) = 1 for all symbols j not appearing in the diagonal of S, i.e., for j E {t, f + 5 , t + 6, . . . , n}, f(j) = 0 for the remaining symbols j . As t is odd, it can be seen from the above list that N s ( j ) is even if and only if j E (t, t + 5, t + 6, . . .,n) and so it folIows that N s ( j )+ f ( j > is

L.D. Andersen

20

odd for all j . We also have that 2(t

+3)

-

n

+ f(j)= [35 ++ ff((jj))

if n = 21 + 1 , if n = 2 t + 3 .

for all symbols j. which shows that Theorem 11 applies. This completes the proof in the case where n is odd. n even. First assume that n = 0 (mod 4) (Hilton's case; what is presented below is essentially his proof). Let t = n/2 and let R be an idempotent partial latin square of side r on the symbols (1,. . . , 1). Extend R to an idempotent partial latin square R' of side t/2 on the same symbols (t/25 r by assumption). We may assume that the diagonal of R' is (1,2,. . . , t/2). By Theorem 3, R ' can be embedded in a latin square T of side t (not necessarily idempotent). Let A be another latin square of side t on the symbols (1,. . . , t}, with the first half of the diagonal being (t/2 + 1, t/2 + 2, . . . , t ) ; the existence of A follows from any one of Theorems 3 , 4 and 8. Let B and C be latin squares of side t on symbols { t + 1 , . . . , 2 t } such that the first half of the diagonal of B is (t + 1, . . . , t + t/2) and the first half of the diagonal of C is (f + t/2 + 1,. . . ,2t). Again, the existence of B and C is clear. Partition T. A, B and C into incomplete latin squares of side t/2 as indicated in Fig. 6.

T:

B:

F] Fl

A:

Fi

Fig. 6.

Let 1, be t h e matrix as shown in Fig. 7. It is straightforward to verify that L is an idempotent latin square of side t i = 7 t . Since R is contained in TI, it is embedded in L. Now suppose that n =- 2 (mod 4). We shall employ a slight modification of the method just described. Let t, R and R' be as before, except that R' is of side ( t - 1)/2. with diagonal (1, . . . , ( t - 1)/2). This time we embed R ' in a latin

21

Embedding latin squares with prescribed diagonal

L:

Fig. 7.

square T of side t (on (1,. . . , f } as before) with the extra requirement that cell ((t + 1)/2, ( t + 1)/2) of T contains the symbol t; this can be done by Corollary 10 (for example), as t = 2(t - 1)/2 + 1 and (t + 1)/2 < t - 1. Again we let A be a latin square of side t on the symbols (1,. . . , t}. We require that the first ( t - 1)/2 places in the diagonal of A are ((t + 1)/2, . . . , t 1) and that cell ((t + 1)/2, (t + 3)/2) of A contains the symbol f. The existence of A is readily deduced from Corollary 10. As before, B and C are latin squares of side t on symbols { I 1,. . . ,2f}. This time we require that the first (t + 1)/2 places in the diagonal of B are (2t, t + 1, t + 2,. . . , t + ( t - 1)/2); B exists by Corollary 10. As regards C, we require that the first (t + 1)/2 places in the diagonal are (t+ (t - 1)/2+ 2 , t + ( t - 1)/2+ 2, . . . ,2t) and that cell (1,2) of C contains the symbol 22; C exists by Corollary 10, as t 2 5 (and (t + 1)/2 < t ) .

+

We then partition the squares in a similar fashion to Fig. 6, but this time T,,

.{

t- 1 2

t-l 2

2

L:

Fig. 8.

t+l 2

t+l 2

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L.D. Andersen

A,, B4and C, are incomplete latin squares of side ( t - 1)/2; T4, A4, B,and C, are incomplete latin squares of side ( t + 1)/2; T2,A2,B3 and C, are incomplete latin rectangles of size ( t - 1)/2 x ( t + 1)/2; and T3, A3, Bz and C2 are incomplete latin rectangles of size ( t + 1)/2 x ( t - 1)/2. Corresponding to Fig. 7 we obtain the latin square L of Fig. 8. It only remains to interchange the symbols t and 2t in the four cells circled in Fig. 8. The latin square L' thus obtained is clearly idempotent, and R is embedded in L'. This completes the proof of Theorem 15. 5. A question of Lindner In [12] Lindner introduced the concept of half-idempotent latin squares in order to prove a result about Steiner Triple Systems. Mainly following Lindner's terminology, a half-idempotent latin square of side 2n on the symbols (1, . . . , 2 n } is a latin square of side 2n on these symbols with diagonal ( 1 , . . . , n, I , . . . , n ) . Lindner is particularly interested in halfidempotent symmerric latin squares; specifically, he says that the half-idempotent symmetric latin square N on (1, . . . , 2 n } contains the half-idempotent symmetric latin square M of side 2m if and only if A4 is on the symbols ( 1 , . . . , m, n + 1. . . . , n + m } and is situated in N as shown in Fig. 9.

Fig. 9.

Lindner proves that if n 2 2 m + 1 and n = m (mod 2), then there exists a half-idempotent symmetric latin square of side 2n containing a half-idempotent symmetric latin square of side 2m. H e proceeds to ask if the same holds for all n and m such that n 3 2m + 1. We show that this is indeed the case. In fact, a somewhat stronger result follows immediately from Theorem 11. We refrain from stating the version of Theorem 11 giving necessary and

Embedding latin squares with prescribed diagonal

23

sufficient conditions for the embeddings of half-idempotent incomplete symmetric latin squares and consider only partial squares. Let us say that M is a half-idempotent parrial symmetric latin square of side 2m on symbols (1, . . . , 2 n } if M is a partial symmetric latin square of side 2m on these symbols whose diagonal is (1, . . . , m, 1 , . . . , m ) . Then we say that M is embedded in the half-idempotent symmetric latin square N of side 2n if (the incomplete square corresponding to) M is situated in N as Fig. 9 indicates. Theorem 16. For any m and any n k 2m + 1 , any half-idempotent partial symmetric latin square of side 2m on symbols ( 1 , . . . , 2 n } can be embedded in a

half-idempotent symmetric latin square of side 2n.

Proof. Let M be a half-idempotent partial symmetric latin square of side 2m on symbols (1, . . . , 2 n } . By filling all empty cells of M (if any) symmetrically, we may obtain a half-idempotent partial symmetric latin square M' of side 2m on the same symbols (i.e., an incomplete square); as 2n - 2(2m - 1 ) 21 4 there is always a symbol available to fill an empty cell. As M' is symmetric and the diagonal is (1,. . . , m, 1 , . . . , m ) , each symbol occurs an even number of times in M'. We then apply Theorem 11 (or Corollary 12) to embed M' in a symmetric latin square N of side 2n on ( 1 , . . . , 2 n } with diagonal (1,. . . , m, 1 , . . . , m, m + 1 , . . . , n, m + 1,.. . , n ) . As both N w ( j ) and f(j) are even for all j , the parity condition is satisfied. Also, each symbol j occurs at least 2(2m)- 2n + f(j) times in M', because 4m - 2n + f(j) s 4m - 2n + 2 =z 0. The situation is illustrated in Fig. 10. It is now a simple matter to rearrange as in Fig. 11, obtaining the required embedding. This proves Theorem 16.

N: Y':

Fig. 10.

El

L.D. Andersen

Corollary 17. For any m and any n 2 2m + 1, there exists a half-idempotent symmetric latin square of side 2n containing a half-idempotent symmetric latin square of side 7 m . Proof. There exists a half-idempotent symmetric latin square of side 2m on symbols (1,. . . , m, n + 1,. . . , n + m } (with diagonal (1,. . . , m, 1, . . . , m ) ; this follows from Theorem 4 (for example)). Now apply Theorem 16 to embed this square in a half-idempotent symmetric latin square N of side 2n on symbols (1,. . . , 2 n } . Corollary 17 was also proved by Hoffman [9]. Lindner wanted this result in connection with a proof of the result of Doyen and Wilson [ 6 ] ,that any Steiner Triple System of order u can be embedded in a Steiner Triple System of order u for any u k 2u + 1, u = 1 or u = 3 (mod 6). We do not present the definitions on this topic here. In [12] Lindner gives a proof of this result which is simpler than the original proof, and he notices that Corollary 17 makes a further shortening possible in the case where u = v = 1 (mod 6 ) .

Corollary 18. A n y Steiner Triple System of order u = 1 (mod 6) can be embedried in a Steiner Triple System of order u for any u z=2v + 1, u = 1 (mod 6). Proof. It suffices to show that there exists a Steiner Triple System of order u with a subsystem of order u, because then the subsystem can be 'unplugged' and replaced by any other Steiner Triple System of order u (on the same elements). Let u = 6n + 1 and let u = 6 m + I , and let N be a half-idempotent symmetric latin square of side 2n on symbols { I , . . . ,2n} containing a half-idempotent symmetric latin square M of side 2m. This exists, by Corollary 17, because

Embedding latin squares with prescribed diagonal

25

u = u = 1 (mod6) and u > 2 u + 1 implies u > 2 u + 5 and so n = ( u - 1 ) / 6 3 (2u + 4)/6 = (12m + 6)/6 = 2 m + 1. Now define a Steiner Triple System S on the set ((1, . . . Zn} x { 1 , 2 , 3 ) )U {w} of u elements by the following rules (Skolem’s construction [12, 151):

{(i, I), ( j , 21, ( j , 3)) E S for all i E (1, . . . , nl , {Q),(i,I), (i - n,2)1, {a,(i,21, (i - 311, 60,(i,3), (i - n, 1)) E s for all j E {n + 1 , . . . , 2 n )

(2)

{(i, (i7 (nijt 2)}, {(i7 2>,( j , 2), (nij, 3)}, {(i, 3), ( j , 3), (nu,1)) E S for all i f j , {i, j } c {I,. . . , 2 n }

(3)

(1)

n 7

and

where nji is the symbol in cell (i, j ) of N. It is a simple matter to check that S is indeed a Steiner Triple System, and that it has a subsystem on the u elements of ((1,. . . , m,n + 1,. . . , n + m } x { 1 , 2 , 3 } )U {m) . This proves Corollary 18.

Acknowledgement

The author is grateful to the referee for drawing his attention to the paper by D.G. Hoffman, and to A.J.W. Hilton for many very helpful discussions on the topic.

Note added in proof

Hilton’s conjecture has recently been proved by L.D. Andersen, A.J.W. Hilton and C.A. Rodger. Matematisk Institut Technical University of Denmark Lyngby, Denmark

References [l] L.D. Andersen, Latin squares and their generalizations, Ph.D. thesis, University of Reading, 1979. [2] L.D. Andersen and A.J.W. Hilton, Generalized latin rectangles XI: Embedding, Discrete Math. 31 (1980) 235-260.

L.D. Andersen

26

(31 L.D. Andersen, R. Haggkvist, A.J.W. Hilton and W.B. Poucher, Embedding incomplete latin squares in latin squares whose diagonal is almost completely prescribed, European J. Combin. 1 (1980) 5-7. (41 J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (MacMillan Press, London, 1976). 15) A.B. Cruse. On embedding incomplete symmetric latin squares, J. Combin. Theory Ser. A 16 (1974) 18-22. (61 J . Doyen and R.M. Wilson, Embeddings of Steiner triple systems, Discrete Math. 5 (1973) 229-239. [7] '1. Evans, Embedding incomplete latin squares, h e r . Math. Monthly 67 (1960) 958-%l.

[8) A.J.W. Hilton, Embedding an incomplete diagonal latin square in a complete diagonal latin square. J. Combin. Theory Ser. A 15 (1973) 121-128. 191 D.G. Hoffman, Completing incomplete commutative latin squares with prescribed diagonals, Auburn University, preprint. [ 10) D. Konig, Theorie der Endlichen und Unendlichen Graphen (Chelsea, New York, 1950). [ 111 C.C. Lindner, Embedding partial idempotent latin squares, J. Combin. Theory Ser. A 10 (1971) 143-245. [I?] C. C . Lindner, A survey of embedding theorems for Steiner systems, in: C.C. Lindner and A. Rosa. eds..Topics on Steiner Systems, Annals of Discrete Mathematics Vol. 7 (North-Holland, Amsterdam, 1980). [I31 C.C. Lindner and T. Evans, Finite embedding theorems for partial designs and algebras, Collection Sminaire de MathCmatiques Sugrieures 56 (Les Presses de I'UniversitC de Montrkal, Montrkal, 1977). { 14 H.J. Ryser, A combinatorial theorem with an application t o Latin rectangles, Proc. Math. Soc. 2 (1951) 550-552.

[IS] T. Skolem. Some remarks on the triple systems of Steiner. Math. Scand. 6 (1958) 273-280.

Annals of Discrete Mathematics 15 (1982) 27-53 @ North-Holland Publishing Company

A DIRECT CONSTRUCTION FOR LATIN SQUARES WITHOUT PROPER SUBSQUARES

Lars D@vlingANDERSEN* and Eric MENDELSOHN** Dedicated to Nathan Mendelsohn on the occasion of his 65th birthday By generalizing a construction due to Katherine Heinrich we produce a latin square of order mp with no proper subsquarefor any positive integer rn and any prime p > 3. Thus the existence of a latin square without proper subsquares is in doubt only for (some) orders n = 2"38.

1. Introduction

The problem investigated in this paper is the following: for what positive integers n does there exist a latin square of order n with n o proper subsquare ? It seems likely that the answer is: for all n # 4,6. We prove here that such a square exists for all n not of the form n = 2"p. Let us remark briefly that the condition that a latin square has no proper subsquare is (much) stronger than the condition that the corresponding quasigroup has n o proper subquasigroup. It is easy to see that there exists a quasigroup of order n without proper subquasigroups for all positive integers n (observed by N.S. Mendelsohn, quoted in [4]). The first papers on the subject were concerned only with latin squares with n o subsquare of order 2, called N2-latin squares. It is easy to check that there is no N2-latin square of order 4 (and none of order 2, obviously). Kotzig, Lindner and Rosa [7] proved that there exists an N2-latin square of any order n which is not a power of 2 (and applied this to construct disjoint Steiner triple systems). McLeish [9] proved that N2-latin squares of order n exist for n = 2" if (Y 3 6. Finally, in [S] Kotzig and Turgeon gave a construction of N,-latin squares of all even orders n f 0 (mod 3) and n+ 3 (mod 5), a class of orders including 24 and 25. They also presented an N2-latin square of order 8 due to Eric Regener. Thus the existence problem for N2-latin squares was completely solved. 'Supported in part by the Danish Natural Science Research Council and Engineering and Natural Sciences Research Council of Canada. ** Supported by Engineering and Natural Sciences Research Council of Canada Operating Grant A7681. 27

L.D. Andersen, E. Mendelsohn

28

It is a general feature of the N,-latin squares constructed in [7,8,9] that they contain (many) proper subsquares of orders different from 2. Thus they are not candidates for a construction for our purpose. In [3] Denniston reported on a computer search for NJatin squares of order 8 (and for latin squares of order 8 with exactly one subsquare of order 2). He found that there are precisely 3 non-isomorphic NJatin squares of order 8, one of them being Regener’s, of course. Since these contain n o subsquare of order 2, they cannot contain a subsquare of order 4. As a proper subsquare of a latin square of order n can have order at most f n (this is very easy to see; it also follows from Ryser’s theorem [ l o ] ) , the only possible order of a proper subsquare of an N2-latin square of order 8 is 3. It turns out that all 3 N-latin squares of order 8 have no subsquare of order 3 and so are completely without subsquares. We reproduce Regener’s square in Fig. 1 . 1

2

3

4

5

6

7

8

2

3

1

5

6

7

8

4

3

1

4

6

7

8

2

5

4

6

8

2

1

3

5

1

5

8

2

7

3

4

6

1

6

5

7

1

0

2

4

3

7

4

5

8

2

1

3

6

8

7

6

3

4

5

1

2

Fig. 1.

On latin squares without proper subsquares not very much is known. The following lemma has been observed by A. Rosa and others; a perfect 1factorization of the complete graph K , on n vertices is a 1-factorization for which the union of any two distinct 1-factors is a Hamiltonian circuit.

Lemma 1. For any integer n, if there exists a perfect 1-factorization of K,, then there exists a latin square of order n - 1 with no proper subsquare. This lemma is really only concerned with even n (K,, for odd n does not even have a 1-factor), and so it can give information about latin squares without proper subsquares only for odd orders. Note that any perfect 1-factorization gives rise to a latin square without proper subsquares, whereas t h e converse is not true. So the perfect 1-factorization is the stronger concept. Perfect I-factorizations of K,, are known to exist when n - 1 is an odd prime [5], when in is a prime [6] and when n is 76, 28, 244 or 344 [l]. In [4] Katherine Heinrich constructed a latin square with no proper subsquare for any order n = pq, where p and q are distinct primes and n # 6. By a

A direct construction for latin squares without p r o p subsquares

29

direct generalization of her construction we prove in this paper that such a square exists of any order n = pm where p is a prime greater than 3 and m is any positive integer; this just means that n must have a prime factor greater than 3, hence the exceptions n = 2"p. A major problem is the exposition of this problem. Although our construction is very similar to that due to Katherine Heinrich, we have tried to adopt a notation more convenient for our purpose. As we deal with latin squares, that is, cells described by two coordinates, as well as their subsquares, and also must deal with products of latin squares, whose entries will be coordinate pairs as well, we choose the following notation. The cell (r, s) of row r and column s of a latin square L will be denoted by (r, s ) ~ ,and its entry will be e((r, s ) ~ ) Thus . in what follows a pair (k,I ) without a subscript denotes a symbol of the base set, not a cell (but presumably an entry of some cell, for example, e((r, s ) ~ = ) (k,I)). We finally leave it to the reader to check that every latin square of order 6 contains a proper subsquare. There is a list of all non-isomorphic latin squares of order 6 in [2]. Throughout the paper we leave out some particularly elaborate cases of the proof corresponding to the case when n = pm is even. These cases present no real additional difficulties but would be very space consuming. 2. The subsquares of a specid product

For any positive integer i we let A(i),B ( i ) and C(i)denote the latin squares of order i, as shown in Fig. 2, where all congruences are mod i. Note that B ( i )is the square B'(i)where e((r, s ) ~ ( , ) )= 3 - ( r + s) with symbols 2 and i interchanged. The purpose of this is to make ail the squares have symbols 1 and 2 in the first two places of the first row (when i > 1).This is very important in the following. We also note that for fixed i, A(i),B ( i ) and C(i)are isotopic, as they are all constant on any set of cells satisfying r + s =constant. (Two latin squares are isotopic if one can be obtained from the other by a permutation of the rows, a permutation of the columns and a relabelling of the symbols.) Lemma 2. If j diuides i and 1 s a 6 j , 1S b d j , then the set of cells {(a + vj, b + p j ) A ( iI )0 d v 6 i / j - 1, 0 d p 6 i / j - 1) forms a subsquare of A(i), and every subsquare of A(i) is of this form.

Proof. It is easy to see that the set of cells forms a subsquare. Its base set of symbols is { a + b - 1 + vj I 0 S v d i / j - 1) (mod i), and it is isotopic to A(i/j). We prove that any subsquare has this form.

L.D. Andersen, E. Mendekohn

B(i) :

1

i

if

e ( ( r , ~ ) ~: ( ~2 ) )i f

r + s :1

r+ s E 3

1

2

i-1

2

i-1

i-2

i-1

i-2

i-3

i

1

2

2

1

i

1

i

i-1

i

i-1

i-2

3

2

1

.--

0

.

..-

3

i

i

1

1

2

4

3

( 3 - (r+s) otherwise

-..-. *.*

4

3

3

2

2

1

5

4

Fig. 2.

Let E be a subsquare of A(i).If E is of order 1 it is of the required form with j = i, so assume that E has order at least 2. Let (a, b)A(i)and (a, b + j ) A ( i ) be the first two cells in the first row of E (to practise the notation: this means ~ (a, b)A(i) and (1,2).~= (a, b + j)A (i) ) .Their entries are, calculated that (1, 1 ) = mod i, a + b - 1 and a + b - 1 + j respectively. Now e((a + j , b)A(i))= a + b - 1 + j , so ( a + j , b)A(i) must belong to E, from which we deduce that (a + j , b + j ) A ( i ) is in E. And e((a + j , b + j ) A ( i ) )= a + b - 1+ 2j = e((a, b + 2j)~(i))(if b + 21 S i), so (a, b + 2j)A(i) is in E. Let k be such that b + kj 4 i and b + ( k + 1)j > i. Then repeated use of the argument above shows first that (4b + kj)A(i)is in E and next that (4b + ( k + 1)j - i ) A ( i )is in E. But b + k j + j - i ~ i + j - i = j < b + j . and so, by the choice of b and j , the cell must in fact be (a, b)A(i). Hence b + kj + j - i = b which implies i / j = k + 1 so that j divides i, and using the same argument on the rows we get that the cells {(a + vj, b + pj)A(i)1 0 v k, 0 p =S k} belong to E. Clearly, by the minimality of j , E contains no other cells. This proves Lemma 2. Corollary 3. if p is a prime, then A ( p ) , B ( p ) and C ( p ) contain no proper subsquare.

For any positive integer m > 1 and any prime p > 3 we now define a latin

A direct construction for Iatin squares without proper subsquares

31

square P ( m ,p ) of order mp on symbols {(i, j ) 1 1S i C m, 1C j e p } . For the remaining part of the paper, when we are concerned with these symbols we always calculate the first coordinate mod m and the second coordinate modp. We shall use a direct product construction with A ( m ) as a frame, each cell of which is substituted by a copy of A @ ) , B@) or C(p). Then, for example, if A @ )is substituted for the cell (r, s)~(,,,), for that copy of A @ ) we will have and

(k,

o,,

= ((r - 1)P + k , (s - 1)P + O P ( m , p )

e(((r- 1 ) +~k , (S - 1 )+~O P ( m , p J = (e((r,s ) A ( ~ ) )e, ( ( k O A = ( r + s - 1, k + 1 - 1).

~ )

Let q be the least prime factor of m. Then P ( m , p ) is obtained by replacing cell (r, s)A(,,,) by a copy of B @ ) if 1CrSmlq

and m - m / q + I C s S m ,

a copy of C@)if

and a copy of A @ ) otherwise. If a cell of A ( m ) is substituted by an A @ ) we say that it is an A@)-cell of A ( m ) , and speak of the copy of A @ ) as the A @ ) of the cell. Similarly for B@)-cells and C@)-cells. If we want to refer to some A @ ) , B @ ) or C@) without specifying which of the three it is, we shall call it a T @ ) ;so we speak, for example, of the T @ )of a cell of A ( m ) . Finally, we shall call the first m - m / q columns of A ( m ) and also the corresponding columns of P ( m , p ) A-columns, and we shall refer t o the remaining cohmns of A ( m ) and P ( m , p ) as mixed columns. The A-area of A ( m ) is the set of cells substituted by A@)’s, and the A-area of P ( m , p ) is the set of cells of all these A@)’s.Similarly for B-area and C-area. Fig. 3 illustrates the definition of P ( m , p ) (in a case where q # 2) and part of the terminology and Fig. 4 depicts P(4,5). In addition to the definition of P ( m ,p ) above, where m > 1, we define P(1,p ) t o be A @ ) for any prime p. The next proposition gives information on subsquares of P(m,p). Before we state it we define a subarray of P(m, p), which is crucial to the next section. Let S ( m , p ) be the m x 2m subarray of P ( m , p ) consisting of the first two cells in the first row of each T @ ) ,that is, S(m, p) consists of the cells shown in Fig. 5 (for brevity we write P ( m , p) = P ) . By the definition of P ( m , p ) each row of S(m. p) contains exactly the symbols (1, I), (1,2), (2, I), (2,219 . . (m,11, (m, 2). 1

L.D. Andersen, E. Mendelsohn

32

C-area __.___

-

-.

__

__

Fig. 3.

The significance of S ( m , p ) in relation to the following proposition is that each proper subsquare of P(m,p) contains a cell from S(m, p) and also, in the same row, a cell not belonging t o S(m, p). This is proved in Corollary 5.

Proposition 4. Let L be a subsquare of P(m,p ) . Then one of the following is true. (a) L consists of a single cell. (b) L is obtained from a subsquare L' of A ( m ) by including the T@)'sof all cells of L (c) For some u for which p I u and u I m,L is a u x u subsquare obtained from a u x u subsquare L' of A ( m ) by including one cell from each T(p) of each cell of L' in such a way that if T@),, denotes the T @ ) to which cell (x, Y ) belongs ~ and i, and j, are such that I.

then, for some d for which gcd@, d ) = 1,

i, = il + (x- l ) d

(modp) (1 G x S u ) .

Let L be a subsquare of P(m, p ) . If the intersection of L and some T ( p ) is not empty, it must be a subsquare of the T @ ) because in the rows and columns of P(m, p) intersecting the T @ )no entries outside the T @ ) have the same first coordinate as that of the entries in the T(p). By Corollary 3 the proof.

A columns

mixed columns

r

P(4.5)

Fig. 4.

CJ W

34

L.D. Andersen, E. Mendelsohn

Fig. 5 .

intersection must be either a single cell or the complete T @ ) .If L contains a complete T @ ) and also some cell outside the T @ ) , it is easy t o see that it contains all of the T @ ) to which the cell belongs. In that case L consists of complete T@)'s, and the corresponding cells of A ( m ) must form a subsquare, so Proposition 4(b) holds. Henceforth we assume that L contains no complete T @ ) ,that is, L contains at most one cell from each T @ ) . It follows that distinct entries of L have distinct first coordinates, and that the cells of A ( m ) for which the corresponding T @ ) intersects L form a subsquare L' of A ( m ) . By Lemma 2 we may assume that L' consists of the cells {(a + vj, b + pji),+,) I 0 S Y S m/j - 1, 0 s p s m / j - 1) for some j , j 1 m, and some a and 6, 1 d a S j , 1s b C j . We put u = m/j. Then L and L' are u x u subsquares. If u = 1 then Proposition 4(a) holds. We now assume that u > 1. Then j C m. For the remaining part of this proof we shall assume that q > 2, where q is the smallest prime factor of m. A few words of explanation about the case q = 2 will be added at the end of the proof. When q > 2, we have that b + j zs m - m/q, because b + j d 2 j d 2 m l q s m - m/q. Thus the first two columns of L are A-columns. For 1 S x s u and 1 S y S u let i, and j, be such that

A direct construction for latin squares withouf proper subsquares

35

that is, row i, of the corresponding T@)’sis used for row n of L, and column j , of the corresponding T@)’s is used for column y of L. Then

+ b - 1, il + j l- 1 ) .

e((l,l),J

= (a

(1)

We also have e((l,2)L)= ( a + b + j - 1, i l + j 2 - l ) ,

e((2, 1)d = ( a + b + j

-

1, i2+ j l - I ) ,

so we deduce that il + j 2 - 1= i 2 + j l - 1 (mod p ) , giving iz= il + (j2- jl) (mod p ) Putting jz - jl = d and repeating the above argument (e((2, 2)L)= e((3, l)L)etc.) we get

ix=il+(n-l)d

(modp)for l s x s u .

(2)

Further, since e((u, 2)L)= (a + ( u - 1)j + b + j - 1, i,

+j 2 -

1)

= ( a + b - 1, i l + ( u - l)d+j2- 1)

we get, from (l),

il + j I- 1 = il + ( u - l)d + j 2 - 1 (m o d p ) , obtaining ud=O

(modp).

(3)

Now Proposition 4(c) will be proved if we prove that gcd(p, d) = 1. Because that will imply, by (3), that p 1 u, which completes the proof of Proposition 4(c). Consider the last column of L‘. Since j s m/q this is a mixed column, and its first cell is a B(p)-cell. The corresponding entry of L is

L.D. Andersen, E. Mendelsohn

36

It has the same first coordinate as e((u, l)'), and so we get, by comparing second coordinates p if i l + j u = l( m o d p ) , 2 if i l + j , , = 3 ( m o d p ) , (modp) 3 - (il + j u ) otherwise . Applying (2) and (3) we get that either and hence il + jl - d = 1 (mod p ) , 3 - il and hence it + jl - d = 3 (mod p ) . or (mod p ) . 4- 2il - j , + d . 1 - il

(4)

We have assumed that q > 2 and so u is odd. We claim that at least one of (i(u + 1). U ) L . and (:(u + 3), u ) is~a C(p>cell. To prove this we must prove that

But this follows from the fact that j

S

m / q and

W e now consider two cases. Case 1. (t(u + I), u ) ~ is ; a C@)-cell.Then, since the entry of ($(u + l), u ) has ~ the same first coordinate as the entry of (t(u - l ) , l ) L (namely a + b + $(u- 3)j - I. mod m),we get from the second coordinates (using (2))

giving (applying (3)) ju = 5 - 2 i , - jl+ 2d

(mod p )

A direct constructionfor latin squares withouf proper subsquares

37

Comparing this to (4)we get: If j,, = 1 - i l (mod p), 1- i l = 5 - i l - ( d + 1 ) + 2d (mod p), implying d=-3

(modp).

(5)

If j,, = 3 - il (mod p), 3 - i l = 5 - i l - (d + 3) + 2d (mod p), implying d=l

(modp).

(6)

Finally, if ju =4- 2il -jl + d (mod p) , 4- 2i1- j l + d = 5 - 2i1- j l + 2d (mod p), implying d=-1

(modp).

(7)

+ 3), u ) ~is. a C@)-cell. In this case e((i(u + 3), u ) ~must ) be the same as e((i(u + l ) , l)L),and so Case 2. &u

i l + i(u - l ) d + jl - 1 =4-

(il+

6(u + l)d + j,,)

(modp),

implying juE5-2i1-jl

(modp).

Again, we compare to (4)and get: If ju = 1 - i, (mod p), 1 - il = 5 - il - ( d (mod p), implying d=3

(modp).

(8)

+ 1) (9)

If j,, = 3 - i l (mod p), 3 - i l = 5 - i l - ( d + 3) (mod p), implying d=-1

d=l

(mo d p ).

( mo d p ).

(10)

(11)

From Cases 1 and 2 we deduce that one of (5)-(ll) holds. As p > 3 we see that in any case, gcd(p, d ) = 1 . This completes the proof in the case q > 2. We omit the proof of the case q = 2, where the B-area and the C-area each

38

L.D. Andersen, E. Mendekohn

take up one quarter of A(m) and together fill the mixed columns completely. In that case the deduction of (2), (3) and (4) is only possible if b + Jzs m - m/q ( = f m ) . If that happens, the proof above works (except that u may be even, in which case cell (iu + 1, u ) is~ a C(p>cell). It is possible, however, that b + j > f m ; this can only happen if u = 2 or u = 3, and a closer analysis of these cases reveals that this cannot occur. We remark that it is possible t o deduce further constraints on p, u, m and j , which are necessary for the situation described in Proposition 4(c). But as we do not need this information here, we shall not present the details. Here we are satisfied with the following corollary of Proposition 4.

Corollary 5. Any proper subsquare of P ( m , p ) intersects S(m, p ) and contains entries not occurring in any cell of S(m, p). h f . Each entry in S ( m , p ) has second coordinate 1 or 2. The statement of the corollary is obviously true for a subsquare satisfying Proposition 4(b). Let L be a subsquare satisfying Proposition 4(c). Since p I u, gcd@, d ) = 1 and i, = i l -k (x - l)d (mod p ) , 1S x S u, i, takes the value of each congruence class m o d p exactly u/p times. Considering the intersection of L and an A-column of P(m,p) it is easy t o see that L certainly contains an entry with second coordinate 1 or 2 (actually both). But in a row of L having i , = 1 (modp) such an entry can only occur in a cell of S(rn,p). Similarly, L contains an entry with second coordinate 3. This proves Corollary 5.

3. Destroying subsquares The object of this section is to modify P ( r n , p ) so as t o destroy all proper subsquares without creating any new ones. We d o this by means of the subarray S ( m . p ) introduced in the last section. We now permute the rows of S ( m , p ) cyclically so that the entries of the second row are placed in the top row, etc., that is, row i + 1 replaces row i for 1 6 i s m - 1, row 1 replaces row m. Only the entries of S(m, p ) are changed, the rest of the cells of P ( m , p ) keep their entries. Let D(rn,p) be the array obtained in this way. As each entry of S(m, p ) is in the same column in D ( m , p ) as in P(m, p), and as each row of S(m, p ) contains the same symbols, D ( m , p ) is a latin square. If m = 1, then D ( m , p ) = P ( m , p ) ; if m # 1, then D(m,p ) # P(m,p). Fig. 6 shows D(4,5), which is obtained from P(4,5) (Fig. 4). The cells permuted are denoted by [a,b].

40

L.D. Andersen, E. Mendekohn

We claim, and this section is devoted to proving, that D ( m , p ) has no proper subsquares.

Lemma 6. No proper subsquare of P ( m , p ) is also a subsquare of D(m.p). Proof. By Corollary 5 each proper subsquare of P ( m , p ) contains a cell of S(m, p) and a row not intersecting S(m, p). Because of this row, if the cells of L were to be a subsquare of D ( m , p ) also, it would be on the same symbols. If m > 1 then. by Proposition 4@) and (c), L does not contain the cell of S(rn,p) below the one known to be in L, and so the entry, in P ( m , p ) , of that cell is not a symbol of L. In D ( m , p ) , however, it is placed in a cell of L, and so the cells of L do not form a subsquare of D ( m , p). Proposition 7 . D ( m , p ) contains no proper subsquare. Proof. Throughout this proof. when we speak of the T @ )of some cell of A ( m ) we shall mean the modified T @ ) of D ( m , p ) , that is, with two new entries in (1, 1 ) ~and ~ ) (1,2)=@, (unless m = 1). Thus the T@)’sare no longer subsquares. By Lemma 6 a proper subsquare of D ( m , p ) would have to include a cell from S(m, p), because otherwise it would also be a subsquare of P ( m , p). We prove that such a subsquare cannot exist. So assume that G is a proper subsquare of D ( m , p). We may also assume that m > 1. Lemma 8. G does not contain two cells in the same T @ ) and not in S(m, p).

Proof. Suppose that G contains two such cells. Then it contains two such cells

with different entries, say el and e2. So G contains the el in the same row as the e2 and the el in the same column as the ez, and also the e2 in the same row as the e l , and the e2 in the same column as the e l , and so on, i.e., we can form a chain of cells containing el and ez which belong to G. Since p is a prime, this chain will visit every row and every column of the T @ ) twice except possibly for the cells of S(m, p ) in the T @ ) ,if any of them were to be included. This implies that the whole of the T @ ) is included in G, also the cells of T @ ) intersecting S(m, p). In D ( m , p), the entries of these cells are also entries in the T@) below the given one, and so we deduce that this T @ )is also included in G. This leads to the conclusion that G = D ( m , p), contradicting that G is a proper subsquare. Hence Lemma 8 is proved.

Lemma 9. G does not contain two cells from S(m, p ) in the same T @ ) .

A direct consrrucn’on for latin squares without proper subsquares

41

Proof. If it did, G would also contain each of the two entries in the column of the other, that is, it would contain two cells from the T @ )below, contradicting Lemma 8.

Lemma 10. G does not contain two cells from S(m, p ) in the same A-column of D ( m , PI. Proof. The entries of S(m, p) all have second coordinate 1 or 2, and two cells of S(m, p) in the same A-column have the same second coordinate. We prove that G cannot contain two such cells with second coordinate 1. The case of second coordinate 2 is similar. Assume that they, in D ( m , p ) , occur in the A(p)’s of cells (al, bl)A(m)and ( ~ 2 ,b l ) A ( , , , ) . Then, in P(m, p) these entries occurred in the A@)’s below, and so they have first coordinates (a1 + 1)+ b11 = al + bl and ( a z + 1)+bl - 1= a2+ bl respectively. Then the entry (a2+bl, 1) in the same row as the (al+ bl, 1) must also belong to G; this entry is in Similarly, the entry S(m, p) and so is in the T@) of (al, bl + a2(al + b171) belonging to S(m, p) and the T @ )of (a2,bl + al - u ~ ) ~ ( is , , ,in) G. We first assume that one of these T@)’s,say that of ( a l , b l + a2 - &)A(,,,), is in an A-column. (Fig. 7 illustrates the situation.)

(a2+bl.l) (a b +a -a ) 1’ 1 2 1 A ( m )

L C-area

Fig. 7.

42

L.D. Andersen, E. Mendekohn

We deduce that G contains a further cell of S ( m , p ) , from the A ( p ) of cell Its entry is (bl + a2 + a2 - a ! ,1). Then al + 61 f bl + a2 + a2 (uz, b, + a2 al (mod m ) , because otherwise P ( m , p ) would contain a 2 X 2 subsquare contained in S ( m , p). But then the (bl + a 2 + a2- a l , 1) in the other column is also in G, belonging to S(m. p) and the A ( p ) of ( a 2 + at- al, bl)A(m).This again leads to a new cell of S ( m , p ) in G, and repeating this argument until we reach a cell containing (al+ b l , 1) in the second column we deduce that G contains entries from t h e two columns corresponding to two columns of a subsquare of A ( m ) . (G may contain further entries from the same two columns, but we only need this first ‘cycle’ for the argument). Because the two columns correspond to the column of a subsquare, G also contains cells from S ( m , p ) in a mixed column, and here it both contains an entry, which was originally in the C-area and an entry which was originally in the B-area. Then, because of the reverse order of the second coordinates for the two entries of S(m, p ) in a C@),in such a mixed column G will contain an entry from the first column of some B(p) and from the second column of some C@) which imply that it will contain the cells from both columns in each of its rows, and so it will contain two cells in S ( m , p ) from the same T(p), which contradicts Lemma 9. Next we assume that (al,61 + u2- a l ) ~ (and ~ ) (a*,bl + at - a t ) A ( m ) both belong t o mixed columns. We first present the proof for the case 9 > 2; the case 9 = 2, which includes some additional difficulties, is done separately afterwards. So assume that 9 > 2. (Fig. 8 illustrates the situation.) Here a2- al f a l - a2 (mod m), because otherwise either P(m,p) would contain a 2 x 2 subsquare contained in S ( m , p ) , or we would be forced to include both cells of S(m, p ) from the same T(p)in G, contradicting Lemma 9. This means that we get two new cells in G by projections, namely from the T@)’sof (al, 61 + a1 - a 2 ) A ( m ) and (a2,bl + az- ~ l ) ~ ( , , , )These . two cells are in S ( m , p ) and their entries have distinct first coordinates, because if they were the same we would have and so

aI+ bl + a , - a2=az+ bl+ a2- al (mod m ) , 3(al - az)= 0 (mod m ),

that is, the ‘distance’ between the two mixed columns of A ( m ) is f m , contradicting the fact that as 9 > 2 there are at most f m mixed columns. Since the two new entries are distinct, each must also occur in G in the opposite row. We claim that at least one of these Occurrences takes place in an A-column. For the entries are in the T@)’sof ( a ~b’)acm) , and (a2, where

A direct constructionfor latin squares without proper subsquares

43

Fig. 8.

a l + b' = a2+ bl+ a2- al (mod m ) , that is,

a2+ b" = al + bl + al - a2 (mod rn ) , bt=b1+2(a2-al)

(mod m ) ,

b"=b1-2(a2-al)

(modm).

Hence b' - bl = 61 - b" = (bl+ a2- al)- (bl+ al - az) (mod m), and since both columns bl + a2- aIand bl + al - az of A(m) are mixed columns, this difference d satisfies (choosing it mod rn so as to take on the smallest possible numerical value) - m/q < d < mlq, implying - f m < d < f m . But then b' and b", each lying within an interval of irn on different sides of bl, cannot both be mixed columns (as m - m/q a i m ) . Hence at least one of b' and b" is an A-column. We also note that b' and b" are distinct columns, because if they were the same we would have 4(az- al)= 0 (mod m ) which is impossible as rn is odd and al f a2 (mod m ) . So we have deduced that G contains at least two A-columns. If the entries of the second of these columns also have second coordinates 1 we are in a situation dealt with in the first part of this proof of Lemma 10. We therefore assume that the entries in the second A-column have second coordinates 2. This means that of the two rows we are looking at one contains entries of

44

L.D. Andersen. E. Mendelsohn

S ( m , p ) originally from the C-area, the other does not. Thus in the mixed columns, where one row contains an entry of S ( m , p) with second coordinate 1, the other contains one with second coordinate 2. Now, as b' # b", G must contain a third row intersecting S ( m , p ) ; because the entry defining 6' must occur in b" and vice versa, and as one of the two columns is an A-column and the two entries both have second coordinates 2, we deduce the existence of a new row intersecting S ( m , p). This third row either contains entries of S(m, p) originally in the C-area, or it does not. In either case it is similar to one of the two rows first considered, in the sense that entries in the same column and in S ( m , p) have t h e same second coordinate. Starting the argument over again with the third row and the one similar to it we eventually reach a contradiction. We finally complete t h e proof of Lemma 10 by considering the case q = 2. The first part of the proof still holds. Additional proof is needed only for the and ) ( a z ,bl + al + a 2 ) A ( m ) both belong to mixed case where (al,bl + a2- a l ) A ( m columns (as illustrated in Fig. 8, except that if q = 2 we know that the B-area and the C-area are larger). These two cells are in different columns, and so each of them implies the inclusion of a cell of the opposite row in G. As in the proof for 4 > 2. we want to argue that the entries of the two new cells have different first coordinates. The argument depended on 4 being at least 3, so here we present a different argument for q = 2. The two cells in question are in the T@)'s of (al, bl + al - a 2 ) A ( m ) and (a?.bl + az- aJA(,,,)respectively; assume that their entries d o have the same first coordinate. Then 3(al- a*)= 0 (mod m).The entries necessarily have the same second coordinate, and we first assume that this second coordinate is 1. This means that they were either both originally in the B-area or both originally in the C-area. Now G also contains the entry (al + bl + al - a2, 1) in the A-column; it is in S(m, p) and in the A @ ) of (a1 + al - a2, b l ) A ( , , , ) . But as the rows of G corresponding to rows al and a2= al - (al- az)of A ( m ) pick up entries of S ( m , p ) both from the B-area or the C-area, and 3(al - a*) 0 (mod m), the row of G corresponding to row al + (al - a2) of A ( m ) must pick up entries of S ( m , p ) from the other area. But then G contains an entry ( a l + b,, Z), contradicting Lemma 9. Next we assume that the entry common t o the two new cells has second coordinate 2. Then (al + b,, 1) also occurs in G in the mixed column corresponding t o column b, + az- al of A ( m ) , and (az+ bl, 1) also occurs in the column of G corresponding to column bl + al - u2 of A ( m ) , and exactly one of these two occurrences will be in a cell of S(m,p), in the row coming from row a: + a2- al of A(m). But if the entry in S(m, p) is (al + bl, 1) this forces G to contain the entry ( a 2 + bl, 2), and if the entry in S(m, p) is (a2+bl, l), G is forced to contain (al + b l ,2). In both cases we obtain a contradiction to Lemma 9.

A direct construction for latin squares withour p r o p subsquares

45

Thus we have proved that also if q = 2 the entries of the cells of G in the T(p)’sof (al, bl + al - a2)A(m)and (a2,bl + u2- &)A(,,,) have distinct first coordinates. Then we can define b’ and b” as in the case q > 2. We have that b’ f b” (mod m ) because otherwise we would get 4(u2- al)= 0 (mod m ) implying that 2(a2- a l )= $n (mod m ) which contradicts that both bl + (a2- al) and 6, ( a 2 - al) are mixed columns. In the case q > 2 we could prove that at least one of b‘ and b” was an A-column. This need not be the case if q = 2, but if it is, the original proof works in this case also. We now assume that both 6’ and bffare mixed columns. Then two new cells of S ( m , p ) are forced into G, namely from the T(p)’sof (al, b”)A(m) and (a2,b’)A(m). The entry of the T ( p ) of (a, bn)A(m) must also occur in the row corresponding to row a2 of A(rn); this happens in a cell of S(m,p) and, it is easy to see, in the column corresponding to column il- 3(az- al) of A(m). Similarly, the entry of (a2,b’)A(m)must occur in the row corresponding to row a , of A ( m ) implying that G intersects row bl + 3(a2- al) of A(m) (calculations mod m, of course). We now prove that at least one of bl 3(a2- al) and bl + 3(a2- al) is an A-column (in fact they are different columns, but we do not need that in the argument). If not, all of columns bl + (a2- al), bl + 2(a2- al), bl + 3(a2- al), bl (a2- ul), bl - 2(a2- al) and bl - 3(a2- al) are mixed columns. We may assume that d = (a2- a l ) is positive, i.e. d E ( 1 , . . . , m - 1). We investigate two cases. Case 1. 1 6 d S j m . Then we get, successively, :m + 1 d bl + d s m, $rn + 1 s bl+2d s m and $m + 1 S b1+3dS m. Similarly, -$m + 1 c bl- d S O , -im + 1 S bl - 2d S O and -$m + 1 s b - 3d S O . From b 1 + 3 d S m and bl-3d2=-$m+1 we get 6 d = ( b l + 3 d ) - ( b 1 - 3 d ) s rn - ( - i m + 1) = i m - 1 and so d Cam. From b l + d z = i m + l and bl-d trn. This is a contradiction. Case 2. irn + 1 d d C m - 1. In thiscase wegettrn + 1 d bl + d s m, m + $rn + 1 d b l + 2 d S 2 m and 2 m + t m + 1 S b l + 3 d S 3 m and also - $ m + l s b , - d S 0, -rn - lm + 1 S bl- 2d S -rn and finally -2m - f r n + 1 s bl - 3d d -2m. From bl + 3d 3 2 m + 1 and bl - 3d S -2rn we get 6d = (bl+ 3d)- (bl- 3d) 3 $m+ 1 +2m = Srn + 1 and so d >im. From b l + d d m and b l - d s - $ m + l we get 2 d = ( b l + d ) - ( b l - d ) s rn - (-im + l ) = f m- 1 and so d
46

L.D. Andersen. E. Mendelrohn

Lemma 11. G does not contain two cells from S(m, p ) in the same mixed column of D(m. P ) . Proof. If it does, each entry will also occur in the other row, and continuing the usual argument it is easy t o see that G contains a cell from S ( m , p ) in an A-column. But then both rows contain such a cell, and we have a contradiction to Lemma 10. Lemma 12. G contains at most one cell from S ( m , p).

Proof. By Lemmas 10 and 11, G cannot contain cells from S ( m , p ) in two different rows. We prove that it cannot contain two cells of S ( m , p ) in the same row. Suppose that it does, say in T@)’s of ( a l ,bJA(,,,) and ( a l ,bz)A(,),say with entries (al+ 61, 7 ) and ( a l+ b2,6 ) . where y and 6 are 1 or 2 (and, for example, if one of the cells is in an A-column, then 6Z y, because otherwise Lemma 10 would be violated). The symbol (al+ bl. y ) also occurs in t h e other column, say in the T @ ) of ( a r .bz),+). Possibly a z = a l . As this occurrence is not in a cell of S ( m , p ) , we get giving

a l + b 1 = a z + b 2 -1 ( m o d m ) , a2=al+ l + b l - b 2 (modm).

This forces into G a cell of the T @ ) of (a2,bJA(,,,), which is not in S ( m , p). The entry of that cell is not ( a l+ b2, 6), because then P(m, p ) would contain a 2 x 2 subsquare with one row contained in S ( m , p). And it cannot have first coordinate al + b2 at all, because then G would contain two entries of that T @ ) ,contradicting Lemma 8. The entry occurs in the other column of G, say in the T @ )of (a?,b2).That Occurrence is not in S(m, p). and so implying

a > +b1- 1 = a 3 - tbz- 1 (mod m )

a3= a l + 1 + 2(b1- 62) (mod m ) .

This again implies the inclusion in G of a cell from the T ( p )of (a3,b1),a cell not in S ( m , p ) . Its first coordinate a 3 +6 , - 1 is not a l + b l . because then the ( a ,+ b,, y ) in the same row would also be included, violating Lemma 8. By continuing this argument we get a sequence of cells of G as indicated in Fig. 9. N o other cells than the first two can be in S(m, p) (by Lemmas 10 and 11); except for the occurrence of ( a l+ b1,y), the entries in the second column never

47

A direct construction for latin squares without p r o p subsquares

a3

\

.

ai+l+2(b -b

1

2

)

(mod. m)

.

projection 7

at :altl+(9.-l)(b

1

-b ) 2

(mod m)

Fig. 9.

have first coordinate al + bt; except if (al+ &, 8 ) occurs in the first column, the entries there never have first coordinate al + b2.The chain may visit the T(p)’s of ( a l ,bZ),+) and ( a l ,bl)A(m) once more, in a row not containing cells of S(m, p), but this does not affect the argument. We have, for all I S 2, al = al

+ 1 + ( I - l)(b,- b2)

(mod m ) .

Clearly the sequence can be continued unless the entry of the T @ ) of

L.D. Andersen. E. Mendekohn

4

(al, b l ) A ( m ) is

giving

(a1 + 62, a), so eventually this must happen. With this 1, we get

a , + 1 + ( / - l ) ( h - 62) + 61- 1

al + b2 (mod m ) ,

I(b1 - b ~=)0 (mod m ) .

We also have that 1 2 3. Now consider the row of G corresponding t o row a2of A ( m ) . The cell of this row containing (a,+b2,S) must belong t o G. It is not in one of the two columns considered so far, say it is in the T @ )of (a2,b),,,). Then giving

a2+ b - 1 = a l + b? (mod m )

~

b = b 2 - ( b 1 - b z ) (modm).

In that column, the entry of the T @ ) of (al,b 2 ) , q m ) also occurs, say in the T @ )of (a, b)A(m).If it does not occur in a cell of S ( m , p ) we get a

implying

+ b - 1 = a / + bZ- 1 (mod m ) ,

a=a,+l

(modm).

But this is impossible. because then the T @ ) of (a,bl)A(m)would have two cells in G, namely one in the first column of G considered, and the (al + bl, 7 ) in the same row, which contradicts Lemma 8. So the entry of the T@)of (a,, bZ)A(m)occurs in the column corresponding t o column b of A ( m ) in a cell of S ( m , p ) , that is, in the row of the other cells of S ( m . p ) in G. With similar arguments we can show that all symbols of G considered so far occur in the first row in cells of S ( m , p ) (find the column containing the entry with first coordinate a, + bl - 1 in the a2-row, and find the entry with first coordinate a, + b2- 1 in that column). This implies that these symbols also form a subsquare of P ( m . p), with one row contained in S(m, p). This contradicts Corollary 5. and SO we have proved Lemma 12. We now finish the proof by proving that G cannot contain just o n e cell from S(m.P). So assume that G contains a cell from S(m, p) in the T @ ) of ( a , ,b l ) A ( m )say , with entry (al + b,, cl), and no other cells from S ( m , p ) ; then any other symbol appearing in G has second coordinate different from 1 and 2. Let ( a l + b 2 1, cz) be another entry of G in the same row, in the T @ ) of ( a l ,b 2 ) A ( m ) . Then b, f b2 (mod m ) , by Lemma 8, and al + b2- I f u I + b1 (mod m ) .

A direct constructionfor latin squares without proper subsquares

49

Now (a1 + b2- 1, c2) occurs in the column of (al+ bl, cl) in a cell not in S(m, p); say it is in the T (p ) of (a2,bl),.,(,,,).Then

so

a2+ b l - 1 = a l + b2- 1 (mod m ) , a2= al + b2- bl (mod m ) .

By the above, a2f al (mod m ) and a2f u1+ 1 (mod m). This forces a cell in the T@) of (a2,b 2 ) A ( m ) into G,say with entry (az+ b2 1, c3). It may be that this entry has first coordinate a l + bl; in that case its second coordinate would have to be cl, by Lemma 8. This can only happen if a 2 + b2- 1= al + bl (mod m ) , giving 2(b2 - bl)

1 (mod m ) ,

and would imply that D(m,p) contains a 2 X 2 subsquare. We refer it to the end of the proof to show that this cannot happen and assume that a2+ 6 2 - 1 f al + bl (mod m ) . Then the entry (a2+ b2- 1, c3) in the first column is also in G, and we may assume that it is not in S(rn,p). Note that it does not belong t o the T @ ) of (a1,bl)A(m), because if it did it would also occur in the first row of that T @ ) (since c3 is not 1 or 2), contradicting Lemma 8. Say that it is in the T @ )of (a37 b l ) A ( m ) ; then

a3= u1+ 2(bz- b l ) (mod m ) . This argument can be continued in the usual way, and we get cells in G from T(p)’sin rows a, = al + ( t - l)(b2 - bl) (mod m ) of A ( m ) . We may assume that we never have to include in G a new entry with second coordinate 1 or 2, and we may also assume that we never get an entry with first coordinate al + bl - 1, because then this entry would appear twice in the T @ ) of (ar,bl).+). Eventually we must get an entry with first coordinate al + bl, say in the T @ ) of (al,b 2 ) A ( m ) . This entry must be (al + bl, cl), because otherwise this and the new entry would both occur in the same T(p).As we get

a1 + b2- 1 = a l + bl (mod m ) l(b2 - bl) = 1 (mod m ) ,

where we first assume that 13 3. The entry (az+ 61 - 1, c2) also occurs in the row of G corresponding to row

L.D. Andersen, E. MendeLFohn

50

a3 of A ( m ) ; the corresponding column of A ( m ) is b, where that is,

a*+ b l - 1 = u 3 + b - 1 (mod m ) . h = bl - (b 2- b,) (mod m ) .

Then G contains a cell in the T@) of (a2,b ) A ( m ) ,and its entry has first coordinate a:!+b - 1

a l + (b2- bl)+ bl - (b2- 61)- 1= a l + b l - 1 (mod m),

which we know is impossible. So it is only left t o consider the case 1 = 2. In that particular case a detailed analysis is necessary, taking into account whether the involved T(p)’s are A@)’s,B@)’s or C@)’s. Now, as 2(bz- b1)= 1 (mod m ) it follows that m is odd and that b2- bl = i ( m + 1) (mod m ) . This implies that q 3 3 and that at least one of columns b1 and b2 is an A-column. And, since a2= al + bz - bl (mod m ) , if row al intersects the B-area then a2 intersects the C-area, and if aZintersects the C-area then al intersects the B-area. We must note that if column bl is a mixed column, then in two particular cases the entry of S ( m , p ) in G has second coordinate different from that usual to an entry in that cell of a T @ ) of that type: if a I= i ( m + 1) the T @ ) of ( a l ,b l ) A ( mis ) an A@),but the entry in G comes from the C @ )below, where second coordinates 1 and 2 are interchanged in the entries of S ( m , p ) , compared to A@);and if al = $(m + 1)+ mlq the T @ )is a C@),but the entries in S(m, p ) come from an A@). We now investigate ail possibilities for the 2 x 2 subsquare with entries in t h e T@)’sof (al,b l ) A ( m ) , ( a ] ,b Z ) A ( m ) . (az, b l ) A ( m ) and (a2, bZ)A(m)- We let i l denote the row used from the T@)’s of (al,bl)A(m)and ( a l ,b 2 ) A ( m ) , i2 the row used from the other two T@)’s,jl the column used from the T@)’s of (al, bl)A(m) and (az,b l ) A ( m )and , j 2 the column used from the remaining two T@)’s.Then, because the cell from G in the T @ )of (al,bl)A(m) is in S ( m , p), il = 1 and jl is 1 or 2. Also, j 2 2 3. The proofs of Cases lb, Ic, Id, le and 2c are left to the reader. Case 1. Column bl is an A-column. Then, obviously, the T@)’s of (ai- ~ I ) A ( ~and ) (az, b l ) A ( m ) are both A@)’s. Case la. The T@)’s of (al, b2)A(m)and (az, bZ)A(,,,) are both A@)’s.Then we have j l -= i ? ’+ j 2 -

1 (modp),

j 2 = i 2 + j l - 1 (modp),

A direct construction for latin squares without proper subsquares

51

implying j 1 - j 2 = i Z - 1 =j2-jl (modp),

which is impossible, because jl # j 2 (mod p). Case lb. The T @ ) of (al,b 2 ) A ( m ) is an A@), that of (al, b Z ) A ( m ) is a B@). Case k. The T @ ) of ( a l , b 2 ) , 4 ( m ) is a B@), that Of (4, b 2 ) A ( m ) is a c@). Case Id. The T @ ) of (al, bZ)A(m) is a C@),that of (4, b Z ) A ( m ) is an A@). Case le. The T @ ) of (al, b 2 ) , 4 ( m ) is a C@),that of (u2,b Z ) A ( m ) is a B@). Case 2. Column bl is a mixed column. Then column b2 is an A-column, and so the T@)’sof (a,,b Z ) A ( m ) and (4, b2)+) are both A@)’s. Case 2a. The T@)’s of ( a l ,b l ) A ( m ) and (a2, b l ) A ( , , , ) are both A@)’s. This is like Case l a unless al = i(m l), in which case the entry in the A @ ) of ( a l , b l ) ~ would ( ~ ) come from a C @ ) ;but if al = t(m + 1), then a2= 1 (mod m), and so Case 2a does not hold. Case 2b. The T @ ) of (a,,b l ) A ( m ) is an A@), that of (a2, bJA(,,,)is a B@). This is only possible if a , = i ( m 1). In that case we get

+

+

j2

p if i2 + jl = 1 (mod p), {3- (i2+ j l ) otherwise , (mod p )

implying 4 - j 1 = i ~ = l - - j 1 (modp) if i 2 + j 1 = 1 (modp),

3 = i2 + j~ + j 2 = 4 (mod p) otherwise , both contradictions. Case 2 ~ The . T @ ) of ( ~ 1 ,bl)A(m) is u B@), that of ( ~ 2 b, l ) A ( m ) is u C@). Case 2d. The T @ ) of (UI,bl)~(,,,) is a C(p), that of (az, b l ) A ( m ) is an A@). This can only happen if al = $(m+ 1)+ mlq, and in that case the congruences become the same as in Case la. Case 2e. The T @ )of ( a l , bl)A(,,,) is a C@),that of (a2, bl)A(,,,)is a B(p). Then al # i(m + 1)+ m/q, and so we get 4 - 0’1 + 1) = i 2 + j 2 - 1 (mod p ) ,

implying

52

L.D. Andersen. E. Mendelsohn

3-jl=iiz- l - - j l

3

(modp) if i z + j l = l ( m o d p ) ,

iz + j l + j z = 3 (mod p ) otherwise

,

in both cases obtaining a contradiction. This completes all cases, and so Proposition 7 is proved.

Theorem 13. For each n not of the form n = 2"3@there exists a latin square of order n with no proper subsquare. Such a square does not exist of orders n = 4 and n = 6 , but it does exist of orders n E (1. 2,8,9,27,81,243}. In particular, when n f 2"3$ and p > 3 is a prime factor of n, then the D(n/p.p ) is of order n and contains no proper subsquare. Proof. Proposition 7 , Lemma 1 and the remarks after it.

4. Final remarks and open questions

Theorem 13 naturally leaves t h e question: does there exist a latin square of order n = 2"3O with no proper subsquare for all (Y and 0, n # 4, n # 6 ? The sporadic orders given in t h e theorem indicates that this may be true. Perhaps the most interesting order to settle the question for would be n = 12. It is possible that by using Regener's Square of order 8 and the perfect one factorization of Klo which gives a subsquare free square of order 9, and two of their isotopes that a similar proof works for 8n, and 9n. It is also likely that a brief exposition of this proof would be at least lo3 pages. Perhaps these cases and the outstanding value n = 12 are best tackled by computer techniques. Another question is: for any n, how many non-isomorphic latin squares of order n without proper subsquares are there ? (Clearly isotopes and conjugates of a square without proper subsquares have no proper subsquares.) In [3] it is reported that the answer for n = 8 is 3, but we d o not know the answer for other n 2 7 . We note, however, that if n has two distinct prime factors p1 and p? both greater than 3, then the two squares D ( n / p l , p l )and D(n/pz,pz) are both without proper subsquares. It is not too hard to check that they are not isomorphic. and so there are at least two non-isomorphic latin squares of order n with n o proper subsquare. Department of Mathematics University of Toronto Toronto. Canada

A direct construction for latin squares without proper subsquares

53

References [ l ] B.A. Anderson, Some perfect I-factorizations, Proc. 7th S.-E. Conf. Conbinatorics, Graph Theory and Computing, Baton Rouge 1976, Congressus Numerantiurn XVII (Utilitas Math., Winnipeg, 1976) pp. 79-91. [2] J. DCnes and A.D. Keedwell, Latin Squares and Their Applications (Academic Press, New York, 1974). [3] R.H.F. Denniston, Remarks on latin squares with no subsquares of order two, Utilitas Math. 13 (1978) 299-302. [4] K. Heinrich, Latin squares with no proper subsquares, J. Combin. Theory Ser. A (1980) 346-353. [S] A. Kotzig, Hamilton graphs and Hamilton circuits, Theory of Graphs and its Applications, Proc. Sympos. Smolenice 1%3 (Nakl. CSAV, 1964) pp. 63-82. 161 A . Kotzig, Groupoids and partitions of complete graphs, Comb. Structures and their A p plications, Proc. Conf. Calgary 1969 (Gordon & Breach, New York, 1970) pp. 215221. [7] A. Kotzig, C.C.Lindner and A. Rosa, Latin squares with no subsquares of order two and disjoint Steiner triple systems, Utilitas Math. 7 (1975) 287-294. [8] A. Kotzig and J. Turgeon, On certain constructions for latin squares with no subsquares of order two, Discrete Math. 16 (1976) 26S270. [9] M. McLeish, On the existence of latin squares with no subsquares of order two, Utilitas Math. 8 (1975) 41-53. [lo] H.J. Ryser, A combinatorial theorem with an application to latin rectangles, Prof. Amer. Math. Soc. 2 (1951) 550-552.

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Annals of Discrete Mathematics 15 (1982) 55-61 @ North-Holland Publishing Company

HIGH CHROMATIC RIGID GRAPHS II

L9sz16 BABAI and Jaroslav NESE-IL

Dedicared to Prof. N.S. Mendelsohn on his 65th birthday An endomorphism of a graph is a map of the vertex set into itself so that edges are mapped onto edges. A graph is rigid if it has no nonidentity endomorphisms. We prove that, given any cardinal K , there exists a rigid graph all edges of which belong to complete subgraphs of power K . Corollaries to this result on certain representations of categories of graphs are derived. We relate boundedness of clique numbers to the problem of extendability to supergraphs of subcategories of graphs.

1. Introduction

For X = (V, E) and Y = (W,F) graphs, a map f : V+ W is a homomorphism if adjacent vertices are mapped to adjacent ones. Endomorphisms of X are homomorphisms of X to itself, They form a monoid (semigroup with identity) End X. The graph X is rigid if End X consists of the identity only. In the theory of full embeddings of categories (see [17]) one proves that any category of algebras is isomorphic to categories of quite particular objects, such as graphs with a given subgraph [3]. The proofs of such results usually depend on the construction of appropriate rigid objects. The difficulty of constructing rigid graphs with large complete subgraphs is indicated by the following trivial observation. Proposition 1.1. A graph X has chromatic number =3if and only if X has a homomorphism to KN (the complete graph on K uertices). Consequently, a rigid graph X containing KR must have chromatic number ch X 2 K + (where K + denotes the cardinal successor of K). The following appears in [3] as Theorem 4.1.

Theorem 1.2. Given a graph X of chromatic number K 2 2, there exists a K +-chromatic rigid graph Y with the following properties:

(i) X is an induced subgraph of Y. (ii) Every triangle of X belongs to Y.

55

L. Bubai, J. Nejeriil

56

We remark that similar results hold for some other categories [2] but there are surprising exceptions [ 1, 10, 111. The purpose of the present note is to construct rigid graphs of chromatic number K' every edge of which belongs to complete subgraphs of power K . We mention that this fact has been used in [12]. Further applications will be given in Section 4.The results generalize those of [3, 4. 6 and 91.

2. Preliminaries Our basic reference is t h e book by Pultr and Trnkov5 [17]. We shall give most of the definitions we need. For graphs X = (V,E) we use 1x1 to denote 1 A clique is a maximal complete subgraph. We call a family of objects mutually rigid if they form a discrete category, i.e., if there are no morphisms other than the identities between them. Theorem 1.2 can be strengthened to yield the following more technical result.

q.

Lemma 2.1. Given a graph X of chromatic number K and an infinite cardinal a 3 1x1+ K ' , rhere exists a family ( Y , :L < a ) of muhtally rigid K'-chromatic graphs such that (i) each Y,contains X as an induced subgraph ; (ii) etlery triangle of each Y,belongs to X ; (iii) 1 Y,lzs a for each L. Proof. The same as the proof of [3, Theorem 4.11 combined with the fact that there exist 2" mutually rigid graphs with a vertices and chromatic number 3. This latter fact follows easily by the method of [7] (cf. [l4]) from the existence of one single rigid digraph of power a [19]. The following notion has been introduced in [9]. Definition. A graph X = (V, E) is called r-completely connected ( r 2 2 an integer), if for any two edges e, e' there is a finite sequence X I , . . . , Xk of complete subgraphs of X such that (i) e E E ( X l )and e' E E ( X k ) ; (ii) for each i, IX, nx.,,) 2 r. We use X @ Y to denote the Zykov sum of the graphs X, Y obtained by taking their disjoint union and adding all edges joining vertices in X to vertices in Y. We denote the supremum of clique sizes in X by cl(X). The subgraph induced by the set of neighbors in X of x E V ( X )will be denoted by X ( x ) .

High chromatic rigid graphs II

57

Lemma 2.2. k t X and Y be connected rigid graphs such that (i) cl(X) 3 r - 2 and cl( Y)3 r - 2; (ii) ch(X(x))+ch(Y)
Proof. The r-complete connectivity follows by observing that for a connated graph X,the sum X @ K,-* is r-completely connected. We have to prove that 2 = X @ Y is rigid. The vertices y of Y are characterized in 2 by ch(Z(y)) 2 ch X > ch(Z(x)) for any x E V ( X ) . Consequently, every endomorphism f of Z sends Y to Y. By the rigidity of Y,f 1 V (Y ) is the identity. For x E V ( X ) ,the vertex f ( x ) has 1 to be a common neighbor of all of V ( Y ) hence f(x)E V ( X ) .It follows that f restricts to an endomorphism of X as well and is therefore the identity. For the purposes of the proof of the main result, we state the following corollary. Corollary 2.3. Let X be a graph with infinite chromatic number K and let a2 + K+. Then there exists u family (X&:L > a ) of mutually rigid K+chromatic graphs with the following properties: (i) X is an induced subgraph of each X,. (ii) Each X, is 3-completely connected. (iii) Jx,~ s a for each L.

1x1

Proof. A combination of Lemmas 2.1 and 2.2. We shall not define here the technical notion of strongly rigid graphs related to the so-called Sip-construction [3,7,9, 741. Clearly, homomorphic images of r-completely connected graphs are r-completely connected. We mention that from this observation it follows that every 2-completely connected rigid graph is strongly rigid.

3. The main result Theorem 3.1. Given a n infinite cardinal K , there exisrs a 2-completely connected K+-chromatic rigid graph Y,every edge of which belongs to a K+-clique. The finite version of this result was proved in [9].

Proof. We construct Y from graphs satisfying Corollary 2.3 by means of amalgamation and a subsequent direct limit. Let us remark that direct limits

L. Babai, J. Neietiil

58

have been systematically used for combinatorial constructions (e.g. in [15, 181). Set X = X x and a = K ' . Let (X(&,,): i < a , n < w ) be a family of mutually rigid K +-chromatic graphs with the properties described in Corollary 2.3. In each X ( , , )fix an edge e(',,)which belongs to a K x . For n < w, we define the graphs Y,, as follows. Let Yo= X(o,o).Set E( Yo)= {e!:L < a} (observing that 1 V(Z)l= IE(Z)l for every infinite rigid graph 2).Let Y,be t h e graph obtained by identifying e? with the edge e(,l)of X ( & ,for ) every L < a ;no other vertices are identified. This way, the graphs Yoand Xfhl) (L < a) have been amalgamated. In a similar fashion, we identify the edges e: of Y,, with e(L,n+l) of X ( L , , +to l) obtain Y , , + l . Finally. let Y = u,,<"Y,,. Except for rigidity, it is trivial to verify that Y has the desired properties. We now prove Y rigid. Let f E End Y. We note that every clique in Y is contained in some X(',,,). The subgraphs X , r . n )intersect each other in s 2 vertices, hence every 3completely connected subgraph of Y is contained in some X(&,,). In particular, the image under f of X ( s n )is contained in some X,,..,.b By mutual rigidity it follows that L' = L, n' = n and f I X(,,,)= id. Consequently, f is the identity map. We remark that the same procedure can be used to prove the following theorem.

Theorem 3.2. Given a connected graph X there exists a 2-completely connected rigid graph Y every edge of which belongs to an induced subgraph isomorphic to X . Moreover, if ch X is infinite one can require ch Y = (ch X ) + .

4. Applications

Applying the Sip-construction [3, 7, 9, 141 the following corollaries are immediate. A functor F : % -+ 9 is called a full embedding if it is one-to-one on objects and for each pair of objects x , y E O b % it is a bijection of Hom,(x, y ) onto Hom9(F(x). F ( y ) ) . A category Y: is binding if the category Gra of graphs with their above defined homomorphisms (and therefore all algebraic categories [7, 81) have a full embedding into %. an infinite cardinal, let Gra(K) be the category of those graphs X of chrurnaric number K' every edge of which belongs to some K x . Then

Corollary 4.1. For

Gra(K) is binding.

K

High chromatic rigid graphs ZZ

59

From Theorem 3.2 it follows that the same holds when an arbitrary connected graph of chromatic number K takes the role of KK in the definition of Gra(K). In particular, given a monoid M and a connected graph X there exists a graph Y such that (i) every edge of Y belongs to a copy of X ; (ii) End Y = M. Let us replace (i) by the weaker condition (if) X is an induced subgraph of Y. The corollary that a graph Y satisfying (i’) and (ii) exists is the result of [3]. The construction given is not very economical in the case when M is a submonoid of EndX. The action of End Y will not extend the action of members of End X. The following notion is designed to express this inefficiency. Definition. Let X be a (not necessarily full) subcategory of Gra. A full embedding F : X+ Gra is called a pseudorealization [16] if the following hold: (i) V ( X ) c V(F(X)) for each object X in X. (ii) The restriction of any homomorphism f : F ( X ) + F ( Y ) to V(X) is a homomorphism f : X + Y which belongs to X and f = F(f). What the second condition says is that the set of X-morphisms X + Y extend uniquely to the full set Hom(F(X), F( Y)).Pseudorealizations are called extension in the stronger sense in [17, p. 441. We shall say that F is a strict pseudorealization if in addition to (i) and (ii), (iii) X is an induced subgraph of F ( X ) ( X E O b X ) . Let now X be a full subcategory of Gra (i.e., for X, Y E O b X , Morx(X, Y) = Morc,.(X, Y)). We say that X is hereditarily strictly pseudorealizable in Gra if every (not necessarily full) subcategory 9 of X admits a strict pseudorealization L?+ Gra. Such categories of graphs admit a handy characterization. Before stating the result, we need an axiom to be added to the usual axioms of set theory: (MI

There is a cardinal y which is greater than any measurable cardinal.

(M) is consistent with ZFC (in fact, the nonexistence of measurable cardinals is consistent with ZFC). The curious role played by (M) in the theory of full embeddings of concrete categories is discussed in detail in [17, Ch. 11.1 (see Theorem 4.3). Theorem 4.2. Under (M), for any full subcategory equivalent :

X of Gra, the following are

60

L. Babai, J. NeSefil

(i) X is hereditarily strictly pseudorealizable. (ii) There exists a cardinal K such that no object in sizes are bounded.)

X contains K R . (Clique

The proof requires the fundamental KuEera-Hedrlin theorem which shows the surprising power of (M). Theorem 4.3 (KuZera-Hedrlin [5, 13; 17, p. 991). Under (M), every concrete category (in particular, every subcategory of Gra) has a pseudorealization in

Gra.

Proof. Assume first that no such K exists. For each cardinal a, let Xu be a graph in X containing K,. Let 9 be the discrete category (the only morphisms are the identities) with O b Y = O b X.Assume there exists a strict pseudorealization F : 9 4 G r a . For some object X , let IF(X)l = a. Then F ( X ) has homomorphisms into K, C X,, C F(X,), hence F is not a full embedding, a contradiction. Assume now that a cardinal K satisfying (ii) exists. Let 2 be any subcategory of X. By the KuEera-Hedrlin theorem there exists a pseudorealization G : X+ Gra. We may assume by [7] that all graphs in the range of G are 3-chromatic. Let ZI and Z z denote two mutually rigid K+-chromatic 2-completely connected graphs. Let zi be a specified vertex of Zi. With each graph X € 2 we associate the graph F ( X ) defined to have vertex set

We define the edge set as follows: the subgraphs induced by {zl} X V ( X )and by V ( G ( X ) )such that they are isomorphic in the natural way to X and G ( X ) ,resp. For each x E V ( X )(x E V ( G ( X ) ) .resp.) the subgraph induced by V ( Z i )x {x} should be isomorphic in the natural way t o Zi ( i = 1,2). Moreover, join (zl, x ) to (z?,x ) by an edge for each x E V ( X ) . (Recall that V ( X ) c V(G(X)).) It is clear how t o turn F into a functor 2 - Gra. Finally, it is a routine matter to check that F is a full embedding and thus clearly a strict pseudorealization.

{zJX

Dept. Algebra Eiitvijs University Budapest, Hungary

KZAA MFF KU Charles University Praha, Czechoslovakia

High chromatic rigid graphs I1

61

References [l] M.E. Adams and J. Sichler, Bounded endomorphisms of lattices of finite height, Canad. J. Math. 29 (1977) 1254-1263. [2] M.E. Adams and J. Sichler, Homomorphisms of bounded lattices with a given sublattice. Arch. Math. 30 (1978) 122-128. [3] L. Babai and J. NeSetiil, High chromatic rigid graphs I, in: A. Hajnal and V.T. %, eds., Combinatorics, Coll. Math. Soc. J. Bolyai 18 (North-Holland, Amsterdam, 1978) pp. 53-60. [4] V. Chvhtal, P. Hell, L. KuEera and J. Neietiil, Every finite graph is a full subgraph of a rigid graph, J. Comb. Theory 11 (1971) 284-286. [5] Z. Hedrlfn, Extension of structures and full embeddings of categories, in: Actes du Congrts Internat. des Mathtmaticiens 1970, tome 1 (Paris, 1971) pp. 319-322. [6] Z. Hedrlfn and E. Mendelsohn, The category of graphs with a given subgraph-with applications to topology and algebra, Canad. J. Math. 21 (1%9) 1506-1517. [7] Z. Hedrlfn and A. Pultr, Symmetric relations (undirected graphs) with given semigroups, Monatsh. Math. 69 (1%5) 318-322. [8] Z. Hedrlfn and A. Pultr, On full embeddings of categories of algebras, Illinois J. Math. 10 (1966) 392-406. [9] P. Hell and J. Neietiil, Graphs and k-societies, Canad. Math. Bull. 13 (1970) 375-381. [lo] J. Kollhr, The category of unary algebras containing a given subalgebra I, Acta Math. Acad. Sci. Hung. 33 (1979) 407417; and 11, ibid. 35 (1980) 5>57. [Ill J. KollL, The category of idempotent 2-unary algebras containing a given subalgebra, in: B. Csikhny and E.T. Schmidt, eds., Universal Algebra, Proc. Conf. Esztergom, Coll. Math. Soc. J. Bolyai 29 worth-Holland, Amsterdam, 1981). [12] V. Koubek, Graphs with given subgraph represent all categories, Comment. Math. Univ. Carolin. 18 (1977) 115-127. [13] L. KuEera, Lectures from the theory of categories, Charles University, 1970, preprint. [14] E. Mendelsohn, On a technique for representing semigroups as endomorphism semigroups of graphs with given properties, Semigroup Forum 4 (1972) 28S294. [15] J. NeSetFil, Infinite precise objects, Math. Slovaca 28 (1978) 25S-260. [16] A. Pultr, On full embeddings of concrete categories with respect to forgetful functors, Comment. Math. Univ. Carolin. 9 (1968) 281-305. [ 171 A. Pultr and Vera Trnkovh, Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories (Academia Praha, Prague, 1980). [18] G. Sabidussi, Existence and structure of self-adjoint graphs, Math. Z. 104 (1968) 257-2W. [ 191 P. Vo p h k a, A. Pultr and Z. Hedrlfn, A rigid relation exists on any set, Comment Math. Univ. Carolin. 6 (1965) 14P-155.

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Annals of Discrete Mathematics 15 (1982) 6Wj8 @ North-Holland Publishing Company

DIRECT CONSTRUCTIONS FOR PERFECT 3-CYCLIC DESIGNS

F.E. BENNETT

Dedicated to Prof. N.S. Mendelsohn on his 65th birthday Let k, A and u be positive integers. A set of k distinct elements {a,, a2, . . . , ~ k is } said to be cyclically ordered by < QZ< 5 3 < . . . < < Q , and the pair ai, &+, are said to be t apart, B)where X is where i + t is taken mod k. A perfect k-cyclic design PD(o, k, A ) is a pair (X, set with 1 x1 = u, called the order of the design, and B is a collection of k cyclically ordered subsets of X,called blocks, such that every ordered pair of elements of X are t apart in exactly A blocks of B for t = 1,2,. . . , k - 1. In this paper, we present simple and direct constructions to show that a PD(u, 3, A) exists whenever the necessary condition Au(u - I)-0 (mod 3) is satisfied, except for the design PD(6,3,1) which does not exist.

1. Introduction

The notion of a perfect cyclic design was introduced by N.S. Mendelsohn [5] and further considerations were given in a subsequent paper [ 2 ] . A perfect k-cyclic design PD(u, k, A ) is a pair (X, B)where X is a set with = u, called the order of the design, and B is a collection of cyclically ordered k-subsets of X, called blocks, such that every ordered pair of elements of X are t apart in exactly A blocks of B for t = 1 , 2 , . . . , k - 1. Note that the block { a , ,u2,. . . , uk} has cyclic order a, < a2< u3< < Uk < ul and the pair Ui, Ui+r are t apart in the block where i + t is taken mod k. This paper is concerned with the construction of perfect 3-cyclic designs and the emphasis will be on the use of simple and direct methods of construction. It should be pointed out that the perfect 3-cyclic designs PD(u, 3 , l ) have been extensively studied since their introduction by Mendelsohn [4]. These designs are more commonly known as cyclic triple systems (CTS) or Mendelsohn triple systems. It is well known that a necessary and sufficient condition for the existence of a PD(u, 3 , l ) is u = 0 or 1 (mod 3), u f 6. The simple constructions presented in this paper show that for all positive integers A and u 2 3 a PD(u, 3, A ) exists whenever the necessary condition Au(u - 1) = 0 (mod 3) is satisfied, except for the design PD(6,3, 1) which does not exist. Some of the constructions given here are reproduced from a previous paper by the author (see [l]) for completeness and the others represent slight modifications of these.

1x1

---

63

F.E. Bentietr

64

2. Preliminaries

If (X, B) is a PD(u, k, A ) , then a simple count shows that the number of blocks in B is A u ( r - l ) / k . This immediately gives us the following theorem. Theorem 2.1. A necessary condition for the existence of a PD(u, k, A ) is A u ( u - 1) = 0 (mod k). The following lemma which is fairly obvious will form a basis for establishing the main result. Lemma 2.2. If a PD(u, k. A I ) and PD(u, k , A2) exist, then there exists a PD(u. k , mh, + nA,) where m and n are non-negative integers. Before proceeding to the main constructions we shall list some examples of small designs for future reference and completeness.

PD(3.3, 1): X = (1.2.3),

B

= {{1,2,3}, (2, 1.3)).

Direct construction for perfecr 3-cyclic designs

65

Remark 2.3. It is known [4] that a PD(6,3, 1) does not exist. However, by Lemma 2.2 and the existence of a PD(6,3,2) and a PD(6,3,3), it follows that a PD(6,3, A ) exists for all A 2.

3. Construction of designs In this section, we demonstrate by the use of direct constructions that a PD(u, 3, A ) exists whenever the necessary condition hu(u - 1)= 0 (mod 3) stipulated in Theorem 2.1 is satisfied, except for the nonexisting design PD(6,3, 1).

Theorem 3.1. A PD(u, 3, 1) exists for all positive integers u where u = 0 or 1 (mod 3), u # 6. Proof. The proof will be divided into four cases. In each case, all subscripts are reduced mod(2t + 1) in the range (0, 1,2, . . . ,2t}. Case 1. u = 6t + 3, where t is a non-negative integer. Let X = {xo, xl, . . . , xz,, yo, y l , . . . , yZh zo, z I , .. . , zZ,} and B be the following collection of cyclic triples:

Then (X, B) is a PD(6r + 3. 3, 1). Cuse 2 . u = 6 t + 1 , where t is X I . . . . . xzr? yo. Y I . . . . , y?,, 20. 2 1 ,. . . cyclic triples:

non-negative integer. Let X = {u,xu, 22,) and B be the following collection of

3

.

Then (X, B) is a PD(6r + 4,3,1). Case 3 . c = 6 t + 6, where t is a positive integer. Let X = { a ,6. c, xu, X L . . . . , x:~, V ~ I .V I . . . . , y,,, zo. ZI,. . . , z Z r }and B be t h e following collection of cyclic triples:

Then (X, B) as a PD(6r + 6 , 3 , 1). Case 4. u = 6t + 7, where t is a non-negative integer. For t = 0, a PD(7,3, 1) is given in the previous section. For t 3 1, let X = {a, 6, c, d, xu, xl, . . . , x2,, yo. y,, . . . Y ? ~ z(], , z I .. . . , z Z r and } B be t h e following collection of cyclic triples:

.

Direct construction for perfecr 3-cyclic designs

67

It is then readily checked that (X, B) is a PD(6t + 7,3,1) and this completes the proof of the theorem.

Theorem 3.2. A PD(v, 3,3) exists for all positive integers v, where u = 2 (mod 3), v35.

Proof. The designs PD(5,3,3) and PD(8,3,3) are exhibited in the previous section. For the remainder of the proof, we shall consider two cases below. In each case, all subscripts are reduced mod(2t + 1) in the range (0, 1, . . . ,2t}. Case 1. v = 6t + 5, where t is a positive integer. Let X = {a, b, xo, xl, . . . , x2,, YO, yl, . . . , y2h 20,21,. . . , ZZJand let (i) B1be the blocks of a PD(5,3,3) based on the set {a, b, xo,yo, zo}, (ii) B2 be the following blocks, each repeated three times:

(iii) B3 be the following blocks, each repeated three times:

B1U B2U B3) is a PD(6r + 5,3,3). Then a direct verification shows that (X, Case 2. v = 6t+ 8, where t is a positive integer. Let X = {a, b, c, d, e, xo, X I , . . . , xzf, YO,YI,. . . , YZ~, 20,21,. . . , z2J and let (i) B1be the blocks of a PD(5,3,3) based on the set {a7b, c, d, e}, (ii) B2 be the following blocks, each repeated three times:

It is then readily verified that (X, B1U B2) is a PD(6t + 8,3,3). This completes the proof of the theorem.

68

F.E. Bemen

If we now combine the results of Theorems 2.1, 3.1 and 3.2 with Remark 2.3 and apply Lemma 2.2, then we readily obtain the following main result.

Theorem 3.3. A necessary and sufficient condition for the existence of a PD(u, 3, A) is Au(u - 1) = 0 (mod 3), except for the nonexisting design PD(6.7, 1). 4. Concluding remark

It is perhaps worth mentioning that slight modifications of the constructions presented in this paper will yield simple and direct constructions for other types of triple systems. These include directed triple systems and, more generally, directed (u. b, r, k, A * ) BIBDs with k = 3 for which recursive methods are employed in [3,6]. Department of Mathematics Mount Saint Vincent University Halifax, Nova Scotia. Canada

References 111 F.E. Bennett. Extended cyclic triple systems, Discrete Math. 24 (1978) 139-146.

121 F.E. Bennett, E. Mendelsohn and N.S. Mendelsohn, Resolvable perfect cyclic designs, J. Combin. Theory Ser. A 29 (1980) 142-1.W. (31 S.H.Y. Hung and N.S. Mendelsohn, Directed triple systems, J. Combin. Theory Ser. A 14 (1973) 310-318. [4] N.S.Mendelsohn, A natural generalization of Steiner triple systems, in: A.O.L. Atkin and B.J. Birch, eds.,Computers in Number Theory (Academic Press, New York, 1971) pp. 323-338. IS] N.S.Mendelsohn. Perfect cyclic designs, Discrete Math. 20 (1977) 6 3 4 8 . (61 J. Sebeny and D. Skillicorn, All directed BIB& with k = 3 exist. J. Combin. Theory Ser. A 29 (19x0) 244-248.

Annals of Discrete Mathematics 15 (1982) 69-80 @ North-Holland Publishing Company

DISTANCE-REGULAR GRAPHS WITH DIAMETER THREE N.L. BIGGS Dedicated to Prof.N.S. Mendelsohn on his 65rh binhday A distance-regulargraph of diameter three is a natural generalization of a strongly regular graph, its structure being determined by five parameters. In this paper the feasibility conditions for such graphs are surveyed, and a classification scheme is set up. Constructions and examples for the various cases are given.

1. Introduction

A graph is said to be strongly regular, with parameters ( k , A, p ) , if (1) each vertex has valency k ; (2) each adjacent pair of vertices has A common neighbours; (3) each non-adjacent pair of vertices has fi common neighbours. Strongly regular graphs have been studied extensively: useful surveys are given by Cameron [4] and Seidel [16]. If the exceptional case p = 0 is excluded, then it is'clear that a strongly regular graph is connected and has diameter two. The natural generalization to graphs with larger diameters is the following, where a denotes the usual distance function. A connected graph with diameter d is said to be distance-regular if there are integers k , bl, bZ,. . . , bd-1, CZ,. . . , cd such that (1) each vertex has valency k ; (2) for each pair (u, u ) of vertices such that a(u, u ) = i there are just bi vertices w for which a(u, w )= i i 1, a(w, u ) = 1 (1 < i d d - 1); (3) for each pair (u, u ) of vertices such that a(u, u ) = i - 1 there are just ci vertices w for which a(u, w )= i - 1, a ( w , u ) = 1 (2 < i d ) . Strongly regular graphs correspond to the case d = 2, with b, = k - A - 1, c2=

p.

The general theory of distance-regular graphs is developed in [l,Chapters 20 and 211. In this paper we shall study the case d = 3 in detail, using some results from the general theory and some special arguments. A mnemonic diagram is very useful (Fig. 1). We put bl = a, bz = p, cz = y, c3= 6, and intend that the diagram should be interpreted in the following way. Given any vertex (the left-hand circle) there are k vertices adjacent t o it (the next circle), and each one of these is adjacent to the initial vertex, to k - CY - 1 69

N.L. Biggs

70

Fig. 1.

vertices in the same circle and to a vertices in the next circle; and so on. In conjunction with the diagram we shall refer to the standard intersection array { k . a,@; 1, y. 6) for the graph. We note that the numbers I and m of vertices at distance 2 and 3 respectively from t h e initial vertex are determined by the equations ka

=

I@

ly.

so that 1 = k a l y , m

=

= ma,

ka@/yS.

2. Feasibility conditions

If we are given an intersection array {k,a,P ; 1, 7, S}, there may be no graph with these parameters; if there is a graph, then certain constraints, known as ‘feasibility conditions’, must be satisfied. We shall begin by collecting together the results from t h e general theory in the form relevant t o our particular case. First, there are feasibility conditions derived from simple counting arguments. (Fl) kculy = 1 and k a p / y S = m are positive integers. (F2) k s a a p and 1 s y S S . (F3) If n = 1 + k + I + m is odd, then k is even; if k is odd, then a is even; if I is odd, then k - /3 - y is even; if m is odd, then k - S is even. More powerful constraints are obtained by applying techniques from linear algebra. We transform the intersection array into a matrix 1

B=

0 Y

a

k-P-Y

B

and we introduce t h e ‘eigenvector’ u = [ l , ul(A), ul(h), u3(A)]‘, defined by treat-

Disfance-regulargraphs with diameter fhree

ing A as an indeterminate and solving Bu

=

71

Ao. Explicitly,

v ~ ( A= ) y-'{A2- ( k - CY - l ) A - k } , vl(A)= A , v3(A)= (76)-'{A3- (2k - (Y - P - - 1)A2+ [ k Z - (a+ P +

+ 2)k

+ (4 + P + y)lA + k 2 - ( P +

Now suppose that a(v, w ) = j (0 S j d 3). The numbers

depend only on h, i, j , not on v and w, and manifestly they are non-negative integers. It turns out that shij is the element in row i and column j of the matrix Bh, where Bo= I, B1= v l ( B )= B, B2= v2(B), B3= v3(B). (The rows and columns of the matrices are labelled 0, 1, 2, 3.) Thus we have the feasibility condition (F4)The elements of v2(B)and v3(B)are non-negative integers. It is a remarkable fact that the eigenvalues of the adjacency matrix of a graph with intersection array { k , a, P ; 1, y, 6) are just the eigenvalues of B.In fact, the eigenvalues are A. = k , and the roots Al, A2, A3 of the equation

The multiplicity of A. (as an eigenvalue of the adjacency matrix) is unity, while the multiplicities of A l , A2, A3 are given by expressions which depend ultimately only on the parameters k, a, P, y, 6. Thus, the integrality of these expressions is a powerful feasibility condition: (F5) If A is a root of (l), then

is a positive integer. The last algebraic feasibility condition is sometimes known as the Krein condition. Let r, s, t be members (not necessarily distinct) of the set {1,2,3) and write vL for vi(Ar). Then we have (F6) The numbers

are non-negative. In Section 4 we shall see that this rather obscure condition does have nontrivial applications.

N.L.Biggs

72

The conditions (Fl)-(F6) are sufficient to eliminate a very high proportion of intersection arrays. However, even if an array satisfies all the conditions there is no certainty that a graph with that array will exist. There is one noteworthy theoretical consequence of these conditions. Theorem 1. A t least one of the eigenvalues A l , A2. h3 must be an integer.

Proof. The formula for m ( h ) given in (F5) has the form nls(A), where s ( h ) is a polynomial with rational coefficients. Now A l , A2, h3 are the roots of a cubic equation (I), say x ( h ) = 0, and we can write s ( h ) = x ( A ) q ( h ) + r(A), where the degree of r(A) is at most 2. It follows that r ( h i )= s(Ai)= n/m(hi). But m ( h i ) is an integer so that the hi’s are, at worst, quadratic irrationals. Since they are the roots of a cubic equation at least one of them must be an integer. This theorem opens up the possibility of a classification in terms of the least integral eigenvalue. However, our classification scheme will follow more obvious, combinatorial. lines.

3. Classification Distance-regular graphs of diameter 3 fall naturally into three classes: antipodal, bipartite, and primitive. We shall say that a graph with intersection array {k,a,p ; 1, y, 6) is antipodal if /3 = 1 and S = k. The name arises from the fact that if u and w are distinct vertices both at distance 3 from a vertex u (that is, ‘antipodal’ to u ) , then v and w are themselves antipodal to each other. This is an immediate consequence of the form of the intersection array.

Theorem 2. Let G be an antipdal distance-regular graph with intersection array { k . a. 1; 1, y, k}. Then a l y is a positive integer m and G is an ( m + 1)-fold covering of the complete graph Kk,I . Proof. The fact that a / y = rn follows from (Fl): rn is just the number of vertices at distance 3 from a given vertex. Define a relation on the vertex-set V of G by setting x y when x = y or a(x, y ) = 3. The remarks preceding the theorem imply that is an equivalence relation on V. Let X denote t h e equivalence class of x, and let denote the set of equivalence classes. Define E as follows:

-

-

{z, >s} E E e

a(x, y ) = I

for some x E i, y E 7.

Distance-regular graphs with diameter three

(e

It is easy to check that E) = is an ( m + 1)-fold covering map.

73

is a complete graph Kktl,and the map x I-+ X

We shall say that a distance-regular graph with intersection array {k, a, B; 1, y, 6) is bipartite if a = k - 1, p + y = k and S = k. This corresponds precisely t o the usual notion of a bipartite graph: If we choose a vertex x and define

P = { y I a(x, y ) = 0 or 21,

B

={

y I d ( x , y ) = 1 or 3},

then the form of the intersection array implies that P and B are the parts of a bipartition.

Theorem 3. Let G be a bipartite graph with intersection array {k, k - 1, k - y ; 1, y, k } , so that the number of vertices of G is n = 2v = 2(k2- k + y ) / y . Let P and B be as above, and define an incidence relation I on P x B by the rule ( p ,b )E I

p and b are adjacent in G .

m e n (P, B, I ) is a symmetric 2-design with parameters (v, k , y).

Proof. Suppose that the vertices in P are called ‘points’ and the vertices in B are

called ‘blocks’. The definition of the parameters implies that each block is incident with k points, and each distinct pair of points is incident with y blocks. Also [PI = IBI = v, so that (P, B, I)is a symmetric 2-design with the stated parameters. We remark that G gives rise to another, dual, design by interchanging P and

B. The two designs may or may not be isomorphic.

From the foregoing definitions we see that the classes of antipodal and bipartite graphs overlap in the case a = k - 1, p = 1, y = k - 1, 6 = k. For each value of k the intersection array {k,k - 1, 1; 1, k - 1, k} is realized by a unique graph Hk, obtained by deleting any 1-factor from the complete bipartite graph K k + l , k + l . This graph is a 2-fold covering of and the associated design is the complete design formed by all k-subsets of a (k + 1)-set. We shall say that a distance-regular graph with diameter 3 is primitiue if it is neither antipodal nor bipartite. The terminology is borrowed from permutation-group theory, for reasons which we shall now explain. A graph G is said to be distance-transitive if, whenever d ( y v ) = a(x, y ) , there is an automorphism g of G such that g ( u ) = x and g ( u ) = y . It is clear that a distance-transitive graph must be distance-regular, but the converse is not true. The main theorem concerning the action of the automorphism group of a distance-transitive graph is due to Smith [MI. In the present context, his result is as follows.

N.L. Biggs

74

Theorem 4. Let G be a distance-transitive graph with diameter 3. If the automorphism group of G acts imprimitively on the uertices, then G is antipodal or bipartite. Theorem 3 justifies the terminology introduced above: a primitive distanceregular graph may not be distance-transitive, but if it is, then the automorphism group acts primitively. In t h e imprimitive case, when the graph is antipodal or bipartite, the existence of a distance-transitive group of automorphisms imposes constraints additional to those which arise from purely combinatorial considerations. We shall encounter examples of this in the following sections.

4. The antipodal case

The general form of t h e intersection array for an antipodal distance-regular graph with diameter 3 is { k , my, 1; 1, y, k}. We shall write r = m + 1, so that the graph is an r-fold covering of and we denote it by ( r Of course, there may be many graphs with the same symbol. The antipodal case was first studied by Gardiner [6] and myself some years ago; more recently, Hoare has also investigated some aspects of it, and I am grateful for his help in the preparation of this section. The case r = 2 is rather special in that we can transform the problem into a more familiar one, and so we shall begin with this case. We have already noted that when r = 2 and y = k - 1 we get the graphs Hk. Suppose that G is a (2 I&+,),, with y f k - 1. Choose a vertex u, and let G i ( v )denote the set of vertices at distance i from v (i = 1,2,3). We see that

and we shall denote the unique member of G 3 ( v )by v ‘ . Now it can be easily verified that if x is in G,(o) then x’ is in G 2 ( v ) , and that X H x’ is an isomorphism of the vertex-subgraphs induced by G,(o) and G2(u).Furthermore. if x and y are in Gl(v),then we find that x and y’ are adjacent

ex

and y are not adjacent.

From these facts it is quite simple to deduce that the subgraph on G,(u)(and on G3(v))is a strongly regular graph with parameters (2p, 2p - Y - 1,p ) in the notation of Section 1, where v = i y and p = i(k - y - 1). Conversely, if a strongly regular graph H of this kind is given then we may construct G in the following way. Let V, V’ be two copies of the vertex-set of

Distance-regular graphs with diameter three

75

H, and u. v’ two additional vertices. The vertex-set of G is defined to be {v, u ‘ } U V U V’, and its edges are (i) {v, x) for all x E V, (ii) {v‘,x’} for all x ’ E V‘, (iii) all edges of H (on V and V’), (iv) {x, y’} whenever x E V , y’ E V’ and (x, y } is not an edge of H. We summarize these results in a theorem. Theorem 5. A n antipodal graph of the form (2 * with y # k - 1 exists if and only if (i) there are integers v and p such that k = 2p + 2 v + 1, y = 2v, and (ii) there is a strongly regular graph with parameters (2p, 2p - v - 1, p ) .

This theorem, proved in its present form by Hoare, had previously occurred in several disguises. We mention in particular the work of Taylor [19] o n regular 2-graphs, and Shult [17]. In our terms the result of Shult may be interpreted as giving a sufficient condition, involving H, for the group of G to act distance-transitively. In this situation, the group of G induces a doubly transitive group acting on the set of antipodal pairs {x, x‘}, and this was the original motivation for the following ‘graph extension theorem’ [ 171. Theorem 6. Let H be a strongly regular graph, v a vertex of H, and Hi ( i = I , 2) the vertex-subgraph induced by the vertices at distance i from v. A sufficient condition for the antipodal graph G constructed as above to be distance-transitive is that there is a permutation h of the vertices of H such that (i) h ( v )= u, is a graph automorphism (i = 1,2), (ii) hi = h 1 (iii) i f x l and xz are vertices of H I and HZrespectively, then h ( x l )and h(xz)are adjacent if and only if x1 and x2 are not adjacent.

The preceding theorems enable us to construct many graphs of the form (2 We might mention in particular two of Cameron’s metrically 6transitive graphs [5]:(2 ehK & ticosahedron, and (2 K&. The corresponding strongly regular graphs are the Paley graphs on 5 and 9 vertices respectively-in other words, the 5-cycle and the line graph of K3,s. We turn now to the case r > 2. It is convenient to begin the analysis by looking at the eigenvalues of a hypothetical ( r * K k + l )We y find that (1) becomes

-

(A

+ 1) ( A 2 -

-

[ k - 1- V I A - k ) = O ,

and so the eigenvalues are -1 and the roots of the quadratic factor, which we

76

N.L. Biggs

shall denote by x, y, choosing x > 0 > y . The formula in (F5) gives the multiplicities: m (- 1) = k and

Suppose first that ry = k - 1, so that the graph has the form ( r . Krl+z)T’ We shall refer to this as the standard type.

Theorem 7 . The intersection array for a graph of standard type satisfies all the feasibility conditions (FI)-(F6) except when r is even and y is odd. Proof. The second part of condition (F3) tells us that if k = ry + 1 is odd, then a = ( r - l ) y must be even. It follows that the array is not feasible when r is even and y is odd. Conversely, it is easy to check that all the feasibility conditions are satisfied, except in the specified case. For instance, the eigenvalues x and y are ?d%, and both have multiplicity $(r - l)(k + 1). If y is even, then k + 1 = ry + 2 is even, while if y is odd, then we have decided that r must be odd, and so r - 1 is even. Thus in both cases the multiplicity is an integer. The existence problem for graphs of standard type, with k equal to a prime power q. was settled by Mathon [14]. A neat version of the construction is the following, due to Neumaier 1151: Let K = GF(q), where q = 1 + ry is a prime power, and let H denote the multiplicative group of non-zero rth powers in K . Define the vertices of the graph to be the ( q 2 - l)/y ‘cosets’ (a , b)H, where a and b are members of K. not both zero, and join (a, b )H to (c, d ) H if and only if ad - bc belongs to H. (Provided r and y are not of the exceptional form mentioned in Theorem 7, - 1 is a member of H.) The graph has a transitive group of autrmorphisms, and it may be verified that it is a graph of standard type (1* Kq+l)r This construction contains the case r = 2 dealt with in Theorem 5, when the associated strongly regular graph is the Paley graph with parameters (2s,s - 1, s) and q = 4s + 1. Another familiar case is (3 K&, the dual of the famous Klein map on a surface of genus 3 1131. for each even When y = 1 the construction yields a graph ( k - 1 * prime power k, and Bondy has shown that such a graph exists only if there is a projective plane PG(2, k). We conclude that there is no (5 * K,),, and the existence of ( 9 . KI1)land (11 * K13)1is still in doubt. For other cases we have no information when k is not a prime power, except that Hoare has shown by exhaustive case arguments that there is no ( 3 . Kith. We now consider the case r > 2, k - 1# ry, and we shall refer to this as the

-

77

Distance-regular graphs with diameter three

sporadic type. Here we have a nontrivial consequence of the Krein condition F6). Theorem 8. If there is a graph of sporadic type ( r * Kk+I)y, then

V S ( k ” - I)(k’”+ Proof. As before, let x > 0 > y, where x and y are the roots of the quadratic A Z - ( k - 1 - ry)h - k = 0. Taking h3= y and computing q333as in (F6), we find

where m = r - 1. Thus if r 2 3, the condition q333 3 0 implies that y 5 -k2”. Using xy = -k, and x + y = k - 1- ry, we deduce the stated condition on r, y, and k .

-

For example, the theorem rules out the possibility of a (3 Kz1)9,aIthough all the other feasibility conditions are satisfied by these parameters. The sporadic cases with r = k (when necessarily y = 1) were investigated by Gardiner [6]. He showed that the existence of ( k Kk+l)lis equivalent to that of a graph with valency k + 1, girth 5 , and (k + 1)2+ 1 vertices-in other words, a Moore graph. It is known [ll] that such graphs exist only when k = 1, 2, 6 , and possibly 56. A doubly-infinite family of graphs of sporadic type has been constructed by Thas [20]:his graphs have the parameters

-

= 2h,

k = 2W

-1

y

= 2(2f-l)h

(kf31).

Apart from the examples just mentioned, no other graphs of the sporadic type are known. In the range k S 13 there are only two unresolved cases: (r, k , y) = (3,8,3)and (5,8,1).

5. The bipartite case Theorem 3 tells us that a bipartite graph with intersection array { k , k - 1, k - y ; 1, y, k } gives rise to a symmetric (u, k , ykdesign with u = ( k 2 - k + y)/y. The converse is quite obvious. All our feasibility conditions are automatically satisfied, provided only that y divides k ( k - l), and this is equivalent to the usual ‘design condition’ Y(U- 1)= k ( k - 1).

N.L. Biggs

78

In this case we have the additional feasibility conditions provided by the results of Bruck, Ryser and Chowla. These conditions assert that if a symmetric ( u , k, ?)-design exists, then (i) if u is even, k - y is a perfect square: (ii) if u is odd, the equation x? =

(k - y ) y ?+ (- l)(u-lNIyz:

has a solution in integers x, y, z, not all zero. It is well known that these conditions rule out the possibility of a symmetric design for many, otherwise acceptable, sets of parameters; for example, there are n o designs with parameters (43, 7, 1) and (22,7,2). It follows that many bipartite arrays, feasible in our sense, cannot be realized by a graph. If a graph of this kind admits a group acting distance-transitively, then the associated design has a group of automorphisms acting doubly transitively on t h e points and blocks. Vijayan [21] pointed out that the symmetric designs satisfying this condition are determined if k or k - y is a prime: The only possibilities are the designs whose points and blocks are the points and hyperplanes of a projective space, and a unique (11,5, ahdesign. Kantor [12, Section 81 gives an extensive survey of doubly transitive designs. A typical result is that if y = 1, then the design, which must be a projective plane, is in fact desarguesian. Non-desarguesian planes d o exist, for example when k = 26, and so they give rise to distance-regular graphs which are not distance-transitive.

6. The primitive case

In this section we shall survey the primitive feasible arrays for small values of k . First we mention some infinite families of arrays and the corresponding graphs. (i) (3t. 2t, 1 ; 1,2, 3). For each t 2 this is realized by the graph whose vertices are the ordered triples of members of a ( t + 1)-set, adjacent vertices corresponding to triples which differ in just one coordinate. (The case I = 1 gives the imprimitive graph H,, in the notation of Section 4.) (ii) (3t,2(t - I). t - 2; 1,4,9}. For each t 2 4 this is realized by the graph whose vertices are the 3-subsets of a (t + 3>set, adjacent vertices corresponding to subsets which intersect in a set of size 2. (When t = 3 we get the antipodal graph (2 + Kd4 .) (iii) {2r, t, 1 ; 1, I , 2). When t is a prime power q, this is realized by the graph whose vertices are the flags of PG(2, q), adjacent vertices corresponding to flags which have a common point or a common line.

Distance-regular graphs with diameter three

79

(iv) {t(t - l), ( t + l)(t - 2), t + 1; 1, 1, t(t - 2)). When t is a prime power q > 2, this is realized by the graph whose vertices are the non-absolute points of a unitary polarity in PG(2, q’), adjacent vertices corresponding to points which lie on the polar line of the other. (The graph is distance-transitive only for q = 3,4.) (v) { s ( t + l), st, st; 1, 1, t + 1). When (s, t ) = (4, q), (q,q3)or (q3,q ) and q is a prime power, these correspond to classical generalized hexagons (see [ 10, p. 3621). The vertices of the graphs are respectively, the points (or lines), and adjacency is defined in points, lines, of the hexagons of type G2, 3D4,3D4, the obvious way. (vi) {s(t2+ t + I), s(t’ + t), st’; 1, t + 1, t2 + t + 1). When q is a prime power and (s, t ) = (q,q2), (q,q), (q3,q2) or (q’, q ) these correspond to classical configurations of symplectic or orthogonal type. For example, in the case s = t = q, the vertices of the graph are the absolute planes of a symplectic polarity of PG(5, q), adjacent vertices corresponding to planes which meet in a line [20]. There are some other known families of graphs of this kind, based on geometrical constructions associated with groups of Lie type. Bannai is compiling a list of distance-regular graphs which contains several such examples. (The classification of all infinite families is a major problem.) Mathon [14] also has some examples, but most of them can be reduced to the ones given here. A list of the primitive arrays with k s 13 is given in Table 1, with the exception of those which belong to one of the families (i)-(vi). In each of these Table 1 Array

n

DR

DT

Comments and references’

4,3,3;1, 1.2 5,4,3;1, 1.2 5,4,2;1, 1,4 6,5,2;1, 1.3 7,6,6;1, 1,2 7,6,5;1,2,3 8,7,5;1, 1,4 10,8,7;1,1,4 10,6,4;1,2,5 11,10,4; 1, 1.5 12,10,2;1,2,8 12,10,3;1,3,8 12,6,5;1, 1,4 13,10,7;1,2,7

35 56 36 57 176 64 135 231 65 210 88

Yes

Yes No Yes Yes No Yes No

Odd graph

?

Yes Yes ?

Yes ?

No

Yes ? ?

68

Yes

175 144

Yes ?

No

Yes No No Yes Yes

04,

unique 17).

[l,p. 1531. PSL(2,19) on cosets of As. ‘Halved cube’ [S] for DT case. Doro; unique DT 18.91.

PI.

Doro, [8]. Doro; unique DT [8]. Line graph of Moore graph [l11.

No

’Where no reference is given, the alleged result is unpublished work of A.D. Gardiner, the present author and others.

80

N.L. Biggs

cases the existence question for a distance-regular (DR) or distance-transitive (DT) graph must be studied by special methods. The feasible arrays were first computed in 1%9, at which time the Krein condition was unknown. In 1974 the computation was repeated using the extra condition. (I am grateful to G.H.J. Meredith for carrying out this work.) The results were published soon afterwards [2] and the list of unresolved cases inspired several people to attack them. The work of Gordon and Levingston 181, Buekenhout and Rowlinson [3], Doro and others has now settled the existence question for all distance-transitive graphs in this range, although there are still some open questions in the distance-regular case. Royal Holloway College University of London Egham. Surrey, UK

References 111 N.L. Biggs. Algebraic Graph Theory (Cambridge University Press, London, 1974). [2] N.L.Biggs, Automorphic graphs and the Krein condition, Geom. Dedicata 5 (1976) 117-127.

(31 F. Buekenhout and P. Rowlinson, The uniqueness of certain automorphic graphs, Geom. Dedicata, to appear. [4] P.J. Cameron, Strongly regular graphs, in: L.W. Beineke and R.J. Wilson, eds., Selected Topics in Graph Theory (Academic Press, New York. 1978). [ S ] P.J. Cameron, Multiple transitivity in graphs, in: R.J. Wilson. ed., Graph Theory and Combinatorics (Pitrnan, 1979). [6] A.D. Gardiner, Antipodal covering graphs, J. Combin. Theory Ser. B 16 (1974) 255-273. 171 A . Gewirtz, Graphs with maximal even girth, Canad. J. Math. 21 (1%9) 915-934. 181 L.M. Gordon and R. Levingston, The construction of some automorphic graphs, Geom.Dedicata. 10 (1981) 261-267. [9] J . I . Hall, Locally Petersen graphs, J. Graph Theory 4 (1980) 173-187. [lo] D.G. Higman. lnvariant relations, coherent configurations and generalized polygons, in: M. Hall and J.H. van l i n t , eds.. Combinatorics (Reidel, Dordrecht, 1975). [ 111 A.J. Hoffman and R.R. Singleton, O n Moore graphs with diameters 2 and 3, IBM J. Res. Dev. 4 (1960) 497-504. [ 121 W.M.Kantor, 2-transitive designs, in: M. Hall and J.H. van Lint. eds., Combinatorics (Reidel. Dordrecht. 1975). [ 131 F. Klein, Uber die Transformationen siebenter Ordnung dex elliptischer Functionen, Math. Ann.

I4 (1879) 428-471. [I41 R. Mathon, 3-class association schemes, Algebraic Aspects of Combinatorics, Congressus Numerantium XI11 (Utilitas Math.. Winnipeg, 1975). [IS] A. Ncumaier. Edge-regular graphs, to appear. [lh) J.J. Seidel. Strongly regular graphs, in: B. BollobAs, ed.. Surveys in Combinatorics (Cambridge University Press, London. 1979). (171 E. Shult. The graph extension theorem, Roc. Amer. Math. Soc. 33 (1972) 278-284. [ 181 D.H. Smith, Primitive and imprimitivegraphs. Quart. J. Math. Oxford Ser. (2) 22 (1971) 551-557. [ I Y ] D.E. Taylor, Regular 2-graphs. Proc. London Math. Soc. 35 (1977) 257-274. [X] J.A. Thas. Two infinite classes of perfect codes in metrically regular graphs, J. Combin. Theory Ser. B 22 (1977) 236-238. ['?I] K.Vijayan. On aclassof distance-transitivegraphs. J. Combin.Theory Ser. B 25(1978) 125-129.

Annals of Discrete Mathematics 15 (1982) 81-95 @ North-Holland Publishing Company

COLOUR SCHEMES Peter J. CAMERON Dedicated to N.S. Mendelsohn on the occasion of his 65th birthday If the k-subsets of a large set are coloured with r colours (all of which are used), then at least r distinct colour schemes of (k + 1)-sets occur. This paper considers the problem of determining colourings with equally many colours and colour schemes. Some general results (including a reduction to r = 2) are given, and the cases k = 2 and (k, r ) = (3,2) are treated in detail. The problem arose from-a question about orbits of permutation groups on unordered sets; applications to permutation groups, and to enumerative model theory, are discussed. Some possible generalisations are mentioned.

1. Introduction

Suppose that the k-element subsets of a set X are coloured with r distinct colours cl,. . . , cn all of which are used. Two (k + 1)-subsets of X are said to have the same colour scheme if they contain the same number of k-subsets of colour ci,for each i. It is known that the number of different colour schemes is at least r, provided that a 2 k + 1. Most of this paper is concerned with the question: for which colourings are the numbers of colours and colour schemes equal? This question can be attacked with a wide variety of techniques, and turns out to be related to problems in group theory, model theory and commutative algebra, as well as combinatorics. Such colourings are of two types: the stable ones, which can occur for infinite (or arbitrarily large finite) sets X; and the others, the unstable colourings. Our question can thus be broken down into two parts: (i) Describe the stable colourings. (ii) Determine how large X must be to ensure that the colouring is stable, and what happens for smaller X. Using Ramsey’s theorem, it is possible to show that the structure of stable colourings is quite restricted. In particular, one can ‘amalgamate’ all but one of the colours so as to obtain a colouring with just two colours and two colour schemes. However, the bound which is obtained for the stable range in this way involves iterated Ramsey numbers, and is quite large. For k S 2 it is possible to

1x1

81

u2

P.J. Cameron

answer both questions completely. We find that, in particular the bound obtained in the Ramsey argument is very far from best possible. One other case can be dealt with completely, that when k = 3 and r = 2. (The importance of the case r = 2 is clear from the preceding paragraph.) Some other general problems about colours are mentioned. The remainder of the paper returns to the more general setting. Applications (and unsolved problems) in group theory and logic are discussed, and we exhibit a connection with graded algebras. The possibility of similar results in other settings is briefly considered.

2. Cdours and d o u r schemes

Given a set X , let (f)denote the set of k-element subsets of X. We assume that elements of (:) are coloured with r distinct colours c I ,. . . , c, and that (in contrast to the situation in Ramsey theory) all of the colours are used. For 1 3 k , the colour scheme of an 1-set L C X is the r-tuple (a,, . . . , ak),where ai is the number of elements of (i) which have colour ci. T h e basic result (whose proof will be outlined) is the following theorem.

Theorem 2.1. Suppose [XI > k + 1, 12 k. Then the number of colour schemes of I-sets is greater than or equal io r, the number of colours of k-sets. Proof. We define t h e colour scheme matrix A of the colouring to be the matrix whose rows are all the colour schemes. The result will follow if we can show that A h a s rank r. Suppose X is finite. Let B be the incidence matrix of I-sets vs. k -sets, with incidence = reverse inclusion. A theorem of Kantor (61 asserts that B has rank (Z), where n = 1x1. Now B can be blocked according t o the partition of rows into colour schemes and columns into colours; and a,, (the ( i , j ) entry of A ) is equal to t h e (constant) row sum of the (i, j ) block B, of B. It follows that A has rank r. The case when X is infinite presents no difficulty, since all colours and colour schemes are represented within a finite subset of X .

Problem 2.2. Given a colouring of the k-subsets of X, let sI be the number of colour schemes of !-sets. Show that SI is a non-decreasing function of 1 for k s I s$Xl. If this problem has an affirmative answer, then the question when equality can hold in Theorem 2.1 can be reduced to the case 1 = k + 1. We shall in any case make this assumption; so, for the next three sections 'colour schemes' will

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83

always be ‘colour schemes of (k + 1)-sets’. If there are equally many colours and colour schemes, then the colour scheme matrix is square and non-singular.

3. Stable cdourings

We call a colour scheme matrix stable if it describes colourings of k-subsets of an infinite set (or, equivalently, of arbitrarily large finite sets); a colouring is called stable if its colour scheme matrix is stable. (It might be preferable to call a colouring stable if it is realised on a subset of an arbitrarily large set with the same colour scheme matrix. I do not know whether this definition is strictly stronger than the one given.) By Theorem 2.1 a stable colour scheme matrix has rank equal to the number of columns; in particular, a stable square colour scheme matrix is non-singular. The most important fact about such matrices is given by the following result. Theorem 3.1. A stable square colour scheme mutrix can be put into lower triangular form by row and column permutations. This theorem is proved in [3]: I outline the argument. We are required to prove the existence of a function f(k, r ) with the property that, if 1x1 3 f ( k , r ) , then any colouring of the k-subsets of X with r colours and r colour schemes has a triangular matrix. Using Ramsey’s theorem, choose f ( k , r ) large enough that there is a monochromatic set of size at least m = max(2k + 1,f ( k , s ) ( s < r),f ( k - 1, t ) ( t S r)}. (Here we are using induction on both k and r.) This ensures that we can remove the points of X one at a time such that, at the first stage when removal of a point destroys a colour or colour scheme, at least m points remain. Then we have

where B is the colour scheme matrix of the colouring remaining after removal of these points. The fact that both A and B have full rank forces B to have equally many rows and columns. Let Y be the set of points remaining, and x the last point removed. Now B is the colour scheme matrix of a colouring of the k-subsets of Y. Also, if we assign to each (k - 1)-subset H of Y the colour of H U{x}, we obtain the colour scheme matrix D or

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P.J. Cameron

according as c = j or not, where c is the row sum vector of C. By induction, both B and D are triangular, whence A is too. I conjecture that the function f can be chosen to be independent of r, the number of colours. Some evidence for this conjecture is given in [ 3 ] . The conjecture can be stated as follows. Problem 3.2. Show that there is a function g ( k ) with the property that, if 1x1 2 g(k), then any colouring of the k-sets of X (with equally many cotours and colour schemes) has triangular colour scheme matrix. One further piece of information can be proved by a similar method. Theorem 3.3. The absolute value of the determinant of a square colour scheme

.

matrix is a product of integers from the set { 1,2, . . , k + 1). with the integer k occurring exactly once in the product.

+1

Proof. Suppose first that, as in the proof of Theorem 3.1,

Then by induction, ldet BI is a product of integers from (1,. . . , k + 11, with k + 1 occurring once only; and either (det DI or kldet Dl is a product of integers from { I , . . . , k}. The result follows. The other possibility is that all but 2k + 1 points can be removed without destroying a colour or a colour scheme; that is, we may assume (XI= 2k + I . Let B be the incidence matrix of k-sets and (k + 1)-sets, as in Theorem 2.1; we arrange the rows and columns so that the ith (k + 1)-set is the complement of the ith k-set. Then (see [l]), B is the adjacency matrix of the odd graph O k + , , and we have

Select from among the k-sets one of each colour; re-order the k-sets so that the chosen ones appear first; then, for each chosen set K,, replace the corresponding basis vector v, by the sum of all the basis vectors corresponding to k-sets of the same colour as K,. The effect of this is to apply elementary column operations to B, which do not change the absolute value of the determinant. The resulting matrix has the property that the first s entries in any row are a colour scheme; so elementary row operations bring it to the form

Colour schemes

85

without changing the determinant. Thus det A divides det B. But k + 1 divides det A, since all row sums of A are equal to k + 1. This proves the theorem. Analogy with Theorem 3.1 and Problem 3.2 suggests the following problem.

Problem 3.4. Show that the determinant of a square colour scheme matrix is bounded by a function of k alone.

4. Colouring edges

In this section we find all stable square colour scheme matrices in the case k = 2, and during this process we give affirmative answers to Problems 3.2 and 3.4 in this case. First, we give the result for k = 1; this is an easy exercise.

Theorem 4.1. If (XI 5 3 and the points of X are coloured with r colours so that r colour schemes of pairs occur, then the colour scheme matrix is one of the following: (i) (2) (only a single colour); (ii)

(iii)

(;

y)

c 1 1) 1 0 1

(all points except one have the first colour);

((XI = 3, all points have different colours).

Thus the conclusion of Problem 3.2 holds for scheme matrix has determinant 2.

1x12 4 ,

and any square colour

The next result gives the complete list of stable colour scheme matrices for k = 2, together with a description of the stable range. It should be regarded as completing the partial result of [4].

Theorem 4.2. Suppose the edges of the complete graph on X (1x13 5 ) are coloured with r colours so that r colour schemes of triangles occur. (i) If 1 x15 9, then the colour scheme matrix is one of the following:

P.J. Cameron

1 2 1 1 1

1

1 1 0

...

(ii) A n y colour scheme matrix not appearing in the list under (i) can be realised on a set X with 1x1 = 5. (iii) The dekrminant of the colour scheme matrix divides 48.

Remark. The actual colourings in (i) are obtained in the course of the proof. Proof. We assume we have a colouring of the edges of the complete graph on X with r colours and r colour schemes, with 1x1 3 5. W e proceed in a series of steps.

i,

S&p 1. If there is a colour occurring in just one colour scheme, then either conclusion (i) holds, or the colour scheme matrix is 2 1 2 0 1 2 0 0 1 0 0 2 1

a

or ( h ) [ i

0 0 1 1 1

the last two occurring on1 if (XIs 6. prod of Step 1. Let u be the colour occurring only in the colour scheme u. Case 1. u = aaa. Then only one colour is used; for every edge meeting an

a-coloured edge itself has colour a. The matrix is of type (c). Case 2. u = aab. Let y z be an a-coloured edge, Y = { x E X I xy has colour b} and 2 = {x E X I x z has colour 6 ) . Then any edge within Y or within 2 has colour 6, while an edge joining a point of Y t o a point of Z has colour a. So only the colours a and b occur. Since 3 5, one of Y,2 contains at least two points, and bbb is a colour scheme; so the matrix is of type (d). Case 3. u = abb. If y t is an a-coloured edge, then all edges xy, xz have

1x1

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87

colour b. So, for any further colour c, bbc is a colour scheme. If bbb is a colour scheme, then t h e matrix is of type (a). Otherwise, no further 6-edges occur, and there is just one colour scheme within X - {y, z}, of type ccc, ccd, or cde. The matrix is thus of type (e), (g) or (h). Moreover, for types (g) and (h), Theorem S 6. 2.1 shows that IX - {y, z}l< 5, or Case 4. u = abc. Let yz be an u-coloured edge, Y = { x I xy is b-coloured} and Z = { x I xz is 6-coloured}. We may assume I YI 2 2. For each colour d of an edge in Y or 2, we have colour schemes bbd and ccd. So, if either set contains a b-coloured edge, then all edges within these sets have colour b; any new colour e between Y and 2 leads to a colour scheme bee, so the colour scheme matrix is o f type (b). In the contrary case, we may suppose all edges within Y or 2 have colour d. Any new colour e between Y and 2 leads to a colour scheme bce and a colour scheme involving d and e ; so there can be no such colour. Similarly we find that none of a, b, c, d is permissible for edges between Y and 2 ; so 2 is empty, and we have type (f).

1x1

Step 2. If (XI2 6 and there is a colour all of whose edges pass through a fixed vertex, then the conclusion of (i) holds. Proof of Step 2. Suppose there are t such colours, t’ of which have valency greater than 1 at the fixed vertex x. Suppose edges of u further colours pass through x, and u = r - t - u. On removing the vertex x, we obtain an edgecolouring with u + u colours and at most t + u + u - (t’ + it(r - 1)+ f t u )colour schemes. (Here equality is only possible if every triangle xyz with xy of one of the t colours and xz of one of the u colours has yz of a different one of the u colours; this implies that u is even.) Theorem 2.1 shows that 1’ + i t ( f + u - 3) =z 0. In particular, t + u 6 3. By Step 1, we may assume that each colour lies in at least two colour schemes. Case 1. t + u = 3. Then t’ = 0 and u is even, so (t, u ) = (1,2) or (3,O). If (r, u ) = (1,2) then x lies on just one edge of colour a ; if b and c colour the remaining edges through x, then abc is the unique colour scheme containing a, contrary t o assumption. If (f, u ) = (3,O) then x lies on just three edges, and 1x1= 4, a contradiction. Case 2. t + u = 2. If t‘ = 1 then t = 2, u = 0. If x lies on one edge o f colour a, then there is a unique colour scheme containing a. So suppose f’= 0. Then w = 0 implies (XI = 3, which is impossible; so u = t = 1. Now the term itu in the inequality can be replaced by tu, and the new bound is met; again there is a colour in just one colour scheme. Case 3. t + u = 1. Then u = 0, r = 1 = t‘, and every edge through x has the same colour. There are u + 1 colours, and 2u colour schemes; so u = 1, and the colour scheme matrix is of type (d).

RB

P.J. Cameron

Step 3. If

1x1 3 6 and the conclusion of

valency at most one at each vertex.

(i) fails, then there is a colour with

Proof of Step 3. Suppose every colour a lies in a colour scheme aaa or aab.

Then every colour scheme is of this form. The average number of colour schemes containing a given colour is at most 2. By Step 1, every colour scheme contains two colours, and every colour lies in two colour schemes. The colour scheme matrix is thus A = 21 + P, where P is a cyclic permutation matrix; det A = 2‘ - (-1)’. By Theorem 3.3, r = 2. Since there are no monochromatic triangles, 1 x1s 5 by Ramsey‘s theorem. Step 4. Proof of the theorem. Suppose the conclusion of (i) fails. Then, removing points one at a time, there is no stage at which a colour disappears when at least five points remain (Step 2); so all colours and colour schemes occur within every 5-set. This proves (ii). Suppose >9; let c be a colour with valency at most 1 at each vertex. If { I , 2,3,4,5} contains two c-coloured edges, say 12 and 34, then the 5-set {1,3,5,6,7} contains only one c-coloured edge. So we may assume {1,2,3,4,5} contains just one c-coloured edge 12. Then we may suppose that 36, 47, 58 are the c-coloured edges in {1,3,4,5,6}, {1.3,4,5,7) and {1,3,4.5,8} respectively, whence {1,3,4,5,9} contains no c-coloured edge, a contradiction. So (i) is proved. Now (iii) follows from (i) and (ii) and the argument of Theorem 3.3 (second case).

1x1

Remark. The numbers 9 and 48 in the theorem are best possible. Example 1. Let X be a set of 8 points, with the structure of affine 3-space over GF(2). Assign a colour to each parallel class of lines of X. There are 7 colours and 7 colour schemes. The colour scheme matrix is the incidence matrix of the Fano plane; its determinant is 24. Example 2. Let X be a set of 5 points. Assign a different colour to each edge of X. There are 10 colours and 10 colour schemes. The colour scheme matrix is the adjacency matrix of the Petersen graph; its determinant is 48. 5. Two cdours: stable cdourings We return to consideration of stable colourings in this section. Theorem 3.1 shows that any stable colour scheme matrix is triangular. If we ‘amalgamate’ all colours except the last, we obtain a stable colouring with two colours and two colour schemes: the colour scheme matrix is

89

Colour schemes

for some i, 1 s i s k . We use the convention that the two colours in this situation are red and blue respectively (so that all-red (k + 1)-sets exist). Empirical evidence suggests dividing the colourings of this form into 1 ree types, illustrated by examples. 1. Simple. There is an i-set I with the property that a k-set is blue if and only if it contains I. 2. Medium. For example, (a) k = (2s + 1)t - 1, i = st, and there is a collection collection F of pairwise disjoint t-sets with the property that a k-set is blue if and only if it contains exactly s members of F; or (b) k = 2i and there is a partition X = Y U Z, Y n 2 = 0, such that a k-set K is blue if and only if ( K n YI = IK n Z ( = i. (In these cases, the structure of the colouring is completely determined by one further parameter, the number of members of <Tin (a), or the cardinality of Y in (b).) 3. Complicated. For example, 3 is a collection of i-sets with the property that two distinct members of 3 intersect in at most 2i - k - 2 points, and a k-set is blue if and only if it contains a (necessarily unique) member of 9. (Here the colouring has a non-trivial combinatorial structure, not in general determined by numerical invariants.) It appears that increasing i makes possible increasing complication of the colouring. Thus, the examples under 3 can only occur for i S i k + 1; if i = k, then the colouring is necessarily of this form. Also, the following theorem is proved in [ 3 ] . Theorem 5.1. Suppose that the k-subsets of a sufficiently large set X are coloured with two colours, having colour scheme matrix

Then, (i) if k > 3i - 1, the colouring is simple; (ii) if k = 3i - 1, the colouring is simple, or medium of type (a) (with s = 1). 6. Two colours: the case k

= 3.

In this section we give a complete description of colourings of triples from a set X, 1x13 7 , with two colours and two colour schemes. We find that all such colourings are stable except for a unique example with = 7.

1x1

P.J. Cameron

90

Consider first the case where there is a red 4-set. Let

be the colour scheme matrix. i = 1: simple, i.e., there is a point x with the property that a triple is blue if and only if it contains x (Theorem 5.1). This is easily proved directly. Let Y be a subset of X with all its triples red, and maximal subject to this. For any x Y, every triple {x, y , y'} ( y , y' E Y ) is blue. If X = Y U { x } we are done; so suppose x' is another point outside Y. Then, for y l , y 2 E Y, exactly one of { y , , x, x'} and { y 2 ,x. x ' } is blue; this is clearly impossible, since I YI > 2. i = 3: a 4-set contains at most one blue triple, that is, the blue triples form a partial Steiner triple system. i = 2: this is the most interesting case. Define a relation on X by the rule that x y if and o n l y if either x = y or no blue triple contains x and y . It is straightforward to check that is an equivalence relation, and that if xi yi ( i = 1,2.3) then xIxzx3 and y l y 2 y 3 have the same colour. Call the colouring reduced if -- is t h e relation of equality. From any colouring we obtain a reduced colouring by choosing one element from each equivalence class. Conversely, given a reduced colouring, we can reconstruct a colouring with equivalence classes of arbitrarily prescribed sizes. So it is enough t o determine the reduced colourings. Now we may no longer assume that 2 7, since by reducing we may decrease A local order is a binary relation T on X with the properties (i) for any x. y E X . exactly one of x T Y , x = y , y T X holds; (ii) for any x E X , the sets { y I X T Y } and ( y 1 Y T X } are linearly ordered by T. For more information on local orders, see [ 3 ] . Note that there is a unique countable dense local order, up to isomorphism. Brouwer has shown that there is a bijection between the isomorphism types of local orders on an n-set and certain binary shift register sequences of length n, namely those in which the bit shifted into the first position is the complement of that shifted out of the last position. The arguments of [3] give t h e following result.

-

-

-

1x1.

-

1x1

Theorem 6.1. A reduced colouring with colour scheme matrix (j ;) on X satisfies one of the following: (i) there is a local order T on X , and rhe blue triples are those cyclically ordered by T : (ii) 1 x1= 6, and the blue triples are the blocks of the unique 2-(6,3,2) design. (The design under (ii) is also known as the unque regular two-graph on 6 points; see [Ill.)

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91

Now consider the possibility that there are no monochromatic 4-sets. Suppose first that (;:) is the colour scheme matrix. Given five points 0, 1, 2, 3, 4, the number of choices of a red triple containing 0 and a quadruple containing it is obviously even. But for each red triple T C {1,2,3,4} there are 0 or 2 red triples in (0)U T containing 0, while for each blue triple in {1,2,3,4}there are 1 or 3 such red triples; so the number of choices is odd, a contradiction. Thus this case cannot occur. Finally, by interchanging colours if necessary, we must consider the colour scheme matrix G:). Take a point 0, and colour the edge xy of the complete graph on X - (0) green or orange according as Oxy is red or blue. Then there is no green triangle. Furthermore, if xyz is an orange triangle, then Oxy, Oyz and Ozx are blue, so xyz is red; thus there cannot be an orange Cclique, and the valency of the green graph is at most 3. If wx, xy, y z and x w are green, then Owx, Oxy, Oyz and Ozw are red, whence wxy, xyz, y z w and zwx are blue, a contradiction. So the green graph has girth at least 5. Suppose u is joined to at least four vertices w, x, y, z by orange edges. Then w, x, y, z contains no monochrome triangle and no green 4-gon. There are just two possibilities: Case 1. wy and xz are green, the other edges orange. We find that uwx and uxy are red, whence wxy is blue; similarly all triples in { w , x, y, z} are blue, a contradiction. Case 2. wx, xy and y z are orange, while wy, xz and w z are green. This time one of wxy and x y z is red, say x y z ; but uxy and uyz are also red, a contradiction. So both green and orange graphs have valency at most 3. Since no regular graph of valency 3 has 7 vertices, we deduce that the number of vertices is 6. Moreover, the only graph of girth at least 5 on 6 vertices with all valencies 2 or 3 is a hexagon. It is now not hard to reconstruct the entire configuration; the red sets form the unique 2-(7,3,2) design,

7. Connections with groups, models, rings

This section outlines, somewhat briefly, some ideas and problems arising in permutation group theory and enumerative model theory which motivated the study of colour schemes of subsets. Let G be a permutation group on an infinite set X , with only finitely many orbits in its action on (F), the set of k-subsets of X. We obtain a colouring of (f) by assigning a colour to each orbit. Then, for 12 k, two I-subsets with different colour schemes necessarily lie in different orbits. So we have

P.J. Cameron

# G-orbits on

(7) 2 #colour

schemes of I-subsets 3 # colours of k-subsets = # G-orbits on

(I).

1x1-

The same argument applies if X is finite, provided that > k + 1 . The inequality for finite groups was proved by Livingstone and Wagner [9], using the character theory of the symmetric group. The argument also shows that if G has equally many orbits o n k-sets and 1-sets, then it also has the same number of orbits on (k + 1)-sets, . . . , ( I - 1)sets, and it acts on a colouring of the kind described in this paper, with equally many colours and colour schemes. In the group case, stronger results can be obtained. They are summarked in the following theorem [3]. Recall that a perniutation group is k-homogeneous if it acts transitively on ( f ) . The preceding argument shows that an 1-homogeneous group is k-homogeneous if f 3 k, 1x1 3 k + 1. This was first proved for infinite groups by Brown [2], by a different method .

Theorem 7.1. Let G be a permutation group on an infinite set X,with equal finite numbers of orbits on (f) and ( t X ) . (i) If G is intransitive then it has a fixed set Y of size at most k and acts k - homogeneously on its complement. (ii) If G is transitive but impnmitive, then it has either two blocks of imprimitivity or blocks of size 2, and the infinite primitive component is (k + 1)homogeneous; moreover, k is even. (iii) If G is primitive but not (k + 1)-homogeneous, then k 3 3 ; moreover, if k = 3, then G is an automorphism group of a local order. with two orbits on (): and two on (:). The following result can be deduced from the first part of the theorem (see 1311.

Theorem 7.2. Let G be a transitive permutation group on an infinite set X . (i) If G has equalfinite numbers of orbits on (f) and (7) with 13 k + 2, then G is I - homogeneous. (ii) Either G is k-homogeneous for every natural number k, or the number of G-orbits on ( f ) is at least ik - c for some constant c. This is the only information known about the sequence (nt),where nk is the number of G-orbits on (f), beyond the bare fact that it is non-decreasing. However, there is some evidence that much more holds in the case where G is primitive (i.e. preserves no non-trivial equivalence relation on X).

Colow schemes

Conjecture 7.3. If G is primitive on X , then either nk faster than any polynomial in k.

93 =

1 for all k, or nk grows

There exists a unique graph r on a countable vertex set, with the properties: (i) any finite graph is an induced subgraph of r; (ii) any isomorphism between induced finite subgraphs of extends to an automorphism of (see [12]). Now if G is the automorphism group of r, then the number nk of orbits of G on k-sets of vertices is obviously equal to the number of isomorphism types of graphs on k vertices. As is well known, this number is at least (and asymptotically equal to) 2k(k-1)'Zlk!; this rapid growth rate may be taken as supporting evidence for the conjecture. And this example is only one of a vast number. It is easy to give sufficient conditions on a class of finite structures for the Examples include existence of such a countable homogeneous structure directed graphs, tournaments, total and partial orders, local orders, graphs with no k-clique, graphs with a fixed bipartition, complete bipartite graphs, circular orders, k -uniform hypergraphs, etc. All the above examples support the conjecture. Clearly its truth would have implications for the enumeration of finite models. These observations have a trivial converse. Given any permutation group G be the relational structure consisting of all on a countable set X, let G-invariant relations on X, and C the class of restrictions of to finite subsets of X. Then r is the countable homogeneous structure corresponding to C. In either the group-theoretic or the model-theoretic situation, it is possible to construct a graded algebra. (i) Let X be an infinite set. Let Vk denote the rational vector space of all functions from (f) to the rationals. Set A = @ v k , and define a multiplication o n A as follows: for f E vk, g E V,, Y E (k.~), set

r

r

r.

r

r

This product is commutative and associative and makes A a graded algebra. Let e denote the constant function in V , with value 1. The argument of Section 2 shows that e is not a zero-divisor. (The matrix representing multiplication by e, as linear transformation from Vk to Vk+l, is the incidence matrix of k-sets and (k + lksets.) (ii) Let G be a group of permutations of X, and V f ,AGthe sets of elements of Vk,A respectively fixed by G. Then AG is a graded subalgebra of A. The dimension of V f is the number of G-orbits on (f), if this number is finite. Now, in a finitely generated graded algebra, the degree of the kth homo-

94

P.J. Cameron

geneous component is bounded by a polynomial in k. So the truth of Conjecture 7.3 would have the following consequence, of interest in its own right: if G is primitive and AG is finitely generated, then G is &-homogeneousfor all k (and AG is the polynomial ring generated by e ) . (iii) Let C be a hereditary class of finite structures: that is, any subset of a member of C carries a substructure which is a member of C. Let Vk be the vector space of functions from isomorphism types of &-element structures in C to the rationals, and A = @ V,,with multiplication as before. Repeating the earlier arguments gives the following result.

Proposition 7.4. Let C be a hereditary class of finite structures for which every C-structure is contained in a larger C-structure. Then the number nk of &-element structures in C ( u p to isomorphism) is a non-decreasing function of k .

In the case where G is the automorphism group of the countable homogeneous structure corresponding to a class C, the algebras under (ii) and (iii) are identical. This approach seems promising, but has not yet contributed anything to the study of groups with equally many orbits on k-sets and ( k + 1)-sets. However, as remarked in [3], such a contribution would follow from an affirmative solution to the following problem. Problem 7.5. Let G be a permutation group on X, with n o finite orbits. Show that e is prime in AG.

The truth of this would also imply that AG is an integral domain under the same hypotheses.

8. Generalisations

Questions about colourings and colour schemes can be posed in more general situations than that of subsets of a set. Consider an ‘incidence structure’ consisting of a set of points and a set of blocks with a relation of incidence between points and blocks. If the points are coloured, then each block has a colour scheme (assuming that the number of points incident with any block is finite). The argument in Theorem 2.1 shows that there are at least as many colour schemes as colours, provided that the rank of the incidence matrix is equal to the number of points. Incidence structures satisfying this condition include balanced incomplete block designs (see Dembowski [ 5 ] ) , e-spaces and f-spaces of finite projective or affine n-space for n 3 e + f + 1 (Kantor [6]),

Colour schemes

95

analogous configurations in finite classical polar spaces (Lehrer [S]), and points and hyperplanes of matroids (Kung [7]). The arguments of this paper clearly suggest considering not one such structure in isolation, but a family of related structures, for example, projective spaces over a fixed finite field. Almost nothing has been done; but, very recently, Saxl [lo] has investigated Steiner triple systems whose automorphism groups have equally many orbits on points and blocks.

Note added in proof (i) Graham (personal communication) has shown the following generalisation of Theorem 3.1: Any stable s x r colour scheme matrix has a lower triangular r x r submatrix with nonzero diagonal. (ii) The answer to Problem 3.4 is negative for k 3 4 . Only the case k = 3 remains open. (iii) MacPherson (personal communication) has verified Conjecture 7.3. Merton College Oxford, England

References [l] N.L. B i a s , Algebraic Graph Theory (Cambridge Univ. Press, London, 1974). [2] M. Brown, Weak n-homogeneity implies weak ( n - 1)-homogeneity, F’roc. Arner. Math. Soc. 10 (1959) 644-647. [3] P.J. Cameron, Orbits of permutation groups on unordered sets I, J. London Math. SOC. (2) 17 (1978) 41M14; 11, ibid., (2) 23 (1981) 249-264. [4] P.J. Cameron, P.M. Neurnann and J. Saxl, An interchange property for finite permutation groups, Bull. London Math. Soc. 11 (1979) 161-169. [5] P. Dembowski, Finite Geometries (Springer, Berlin, 1W). [6] W.M. Kantor, On incidence matrices of finite projective and affine spaces, Math. Z. 124 (1972) 3 15-3 18. (71 J.P.S. Kung, On the Radon transform of a combinatorial geometry, J. Combin. Theory Ser. A 26 (1979) 97-102. [8] G.I. Lehrer, On incidence structures in finite classical groups. Math. 2. 147 (1976) 287-299. [9] D. Livingstone and A. Wagner, Transitivity of finite permutation groups on unordered sets, Math. Z. 90 (1965) 393-403. [ 101 J. Saxl, On points and triples in Steiner triple systems, to appear. [ l l ] J.J. Seidel, A survey of two-graphs, Colloq. Int. sulle n e o r i e Combinatorie (Accad. Naz. Lincei, Roma, 1977) pp. 481-511. [ 121 R.E. Woodrow, There are four countable ultrahomogeneous graphs without triangles, J. Combin. Theory Ser. B 27 (1979) 168-179.

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Annals of Discrete Mathematics 15 (1982) 97-103 @ North-Holland Publishing Company

THE ANALYSIS OF DIRECTED TRIPLE SYSTEMS BY REFINEMENT Marlene J. COLBOURN and Charles J. COLBOURN Dedicated to N.S. Mendelsohn on the occasion of his 65th birthday An invariant for deciding isomorphism of directed triple systems is developed by exploiting the asymmetry of blocks. This invariant can be computed efficiently, and for small systems usually succeeds in determining isomorphism. It is shown, however, that for all orders which are 1 (mod3). there exist systems for which the invariant gives no information regarding isomorphism.

1. Introduction

Directed triple systems (DTS) were introduced by Hung and Mendelsohn [ 131 as a generalization of Steiner triple systems (STS). Existence of DTS has been settled [13]; enumeration, however, remains open. In the case of Steiner triple systems, two types of enumeration results are known. Asymptotic bounds are known on the number of nonisomorphic STS. In addition, the number of nonisomorphic STS has been determined exactly for small orders. This latter type of result requires primarily a practical method for determining whether two designs are isomorphic. For balanced incomplete block designs in general, no algorithm is known to solve this problem in less than exponential time; in fact, such an algorithm will likely be hard to find since deciding isomorphism of block designs is polynomial time equivalent to deciding graph isomorphism [7]. The problem for DTS is not this general, however. In fact, there is a subexponential time isomorphism test for STS [6,9,16]. This method for STS generalizes t o many classes or 1-factorizations IS]; further, it is easy t o see that it yields a subexponential time isomorphism test for DTS. Although straightforward, this is somewhat surprising since no such result is known for triple systems with h = 2. We prefer an isomorphism test which guarantees polynomial running time. Short of achieving this goal, we may consider techniques which sometimes decide isomorphism. Selecting any feature of a design which is invariant under isomorphism, we compare two designs with respect to this feature. If they differ, they are nonisomorphic; the converse does not hold in general. 91

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M.J. Colbown. C.J. Colboum

One wants an invariant which can be easily and efficiently computed (i.e., in polynomial time), and succeeds in distinguishing nonisomorphic designs. For DTS, we can adapt many of the invariants developed for STS [9], by exploiting the similarities between DTS and STS. This is a relatively straightforward process. Instead, we examine a more interesting process of exploiting the differences between STS and DTS. In so doing, we adapt a very useful graph invariant t o handle designs.

2. Refinement

A directed triple system of order v is a u-set V of elements together with a collection B of 3-tuples of elements, called blocks; each ordered pair of elements appears in precisely one block in B (a block (x, y, z) contains the pairs (x, y ) , (x, z ) , and (y, z)). Directed triple systems differ from STS primarily in that the blocks are ordered. This ordering provides an immediate isomorphism invariant. Given a DTS, we define, for each element x, a position uector p(x) = (cl(x), cz(x), CJ(X)). where c,(x) is the number of times x appears in position i of a block. The position vector of an element is invariant under isomorphism. This invariant is comparable to vertex degrees in directed graphs; an invariant for a vertex x is a pair consisting of its indegree and its outdegree. We can profit by carrying this analogy with digraphs further. In a digraph, two vertices with different degrees are non-similar. Of course, the converse does not hold. If the degrees of the neighbours of a vertex x differ from the degrees of the neighbours of another vertex y , x and y are non-similar. This is based on a trivial but important observation: an isomorphism carrying x t o y carries the neighbourhood of x to the neighbourhood of y. Using this simple fact leads to an isomorphism technique called refinement [4, 181. In polynomial time, refinement decides isomorphism for almost all graphs [ 11. For graphs, then, we rarely have to appeal to an isomorphism method which does not guarantee polynomial running time. We introduce refinement formally for directed tripie systems. We define a sequence of labels qj for the blocks, and a sequence of labels pi for the elements. Initially, we set po(x) = p(x)-i.e., an element’s initial label is its position vector. For a block b = (x, y , z), we define q,+l(b)= (pl(x), p j ( y ) ,pi(,?)). Finally, we define p j ( x )= ( c l ( x ) ,cZ(x), c3(x)), where c j ( x ) is the multiset {q,(b)I x appears in position j of 6 ) . Generalizing our trivial observation about neighbourhoods’ we conclude that for each i, pl(x) is an invariant of element x. One difficulty with using such refinement invariants is that computation time increases as a function of i. It is therefore essential to observe that, as with graphs [4], for a DTS of order u, we obtain no new information by considering

The analysis of directed triple systems

99

the labels pi for i 3 u. This still does not make the computation feasible-the label pi(x) has length which is exponential in i. To circumvent this, after the pi labels are computed, we sort them (omitting duplicates) and replace each label by its rank in this sorted list. This keeps the labels short, and guarantees that the pu labels are computed in polynomial time [lo]. Now suppose that we have two DTS DI and Dz for which we have computed pu labels. Of the u ! mappings carrying DI to D2, the only candidates for isomorphisms carry an element x to an element y only if pu(x)= pu(y).Having computed the pu labels, the remaining candidates for isomorphisms number at most u ! . This is realized when all of the pu labels are equal, i.e., the pu do not partition the elements at all. In this case, we say that refinement fails completely. At the other extreme, all of the pu labels are different, i.e., the py labels partition the elements into singletons. The latter case is definitely the preferred. Here, there remains only one candidate mapping for isomorphism. To decide isomorphism, we need only decide if this one candidate is indeed an isomorphism. In practice, a more sophisticated approach is implemented, using the fact that the sorted lists of pi labels for each i comprise a design invariant. Often this immediately rules out the possibility of isomorphism. We remarked earlier that refinement leads to a polynomial time algorithm for deciding isomorphism of almost all graphs. No such theoretical result is known for DTS. In practice, we found that for small orders, refinement was easily and quickly carried out, and has a good record in deciding isomorphism of DTS. Refinement is a complete success when the pu labels partition the element set into singletons; moreover, the more classes produced in this partitioning, the easier it is to complete the task with other (guaranteed) methods [18]. There are 32 nonisomorphic DTS of order 6. Refinement produced partitions consisting of six classes (i.e., singletons) for 26 of these designs, three classes for 4, and two classes for 2. It did not fail completely on any DTS(6). There are 2368 DTS of order 7. Refinement produces seven classes (singletons) for 1960 of these; at the other extreme, it produced only one class (i.e., failed completely) for 271 designs. In the remaining cases, it produced six classes for 6 DTS, five classes for 7, four classes for 100, and three classes for 24. At least for small orders, we conclude that refinement is a powerful tool for deciding isomorphism for DTS, as it is for graphs.

3. Regular directed triple systems Refinement of DTS typically results in a nontrivial partitioning of the element set. Even when it fails to produce a partitioning into singletons, any nontrivial partitioning can be used t o assist in the determination of isomor-

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M.J.Colboum, C.J. Colboum

phism [MI. One concern, then, is to decide when refinement fails completely. If any two elements have different position vectors, a nontrivial partitioning will be obtained. Refinement therefore fails completely if and only if the DTS is regular, i.e., every element has the same position vector. This repeats the analogy with graphs, where refinement fails completely on regular graphs. Regular graphs of every order exist; by contrast, we have the following lemma. Lemma 3.1. A necessary condition for a regular DTS(u) is u = 1 (mod 3). Proof. A DTS is regular only if every element's position vector consists of three equal integers. Each element appears u - 1 times, whepce u - 1 = 0 (mod 3).

This simple lemma has a surprising corollary, that refinement never fails completely for a DTS(3m). In the remainder of this section we construct regular DTS(u) for u = 1 (mod 3). Lemma 3.2. A regular DTS(u) exists for u = 1 , 4 (mod 12).

Proof. Suppose u = 1, 4 (mod 12). Let (V, B ) be an S(2,4, u ) , whose existence follows from [I 11. Replace each block b = {w,x , y , z } of B by ( w , x, y), ( y , z, x ) , (x, w, z), and (2,y , w)-in other words, replace a block by a copy of a regular DTS(4). The result is a regular DTS(u). An equally simple construction shows the following. Lemma 3.3. A regular DTS(u) exists for u = 1 (mod 6).

M. Suppose u = 1 (mod 6). Let {(xi.y,, zi)I 1 S i S ( u - 1)/6} be a set of starter blocks [9] for a cyclic STS(u), whose existence follows from [ 171. Then ((xi, yi,I,), (z,, y,, x i ) I 1S i S ( u - 1)/6} is a set of starter blocks for a DTS(u). Further, it is regular. since it is cyclic. This suggests one method of constructing regular DTS(u)--construct ones. Before examining this, note the following lemma.

cyclic

Lemma 3.4 [ S ] . Triple systems with A = 2 having a cyclic autornorphisrn exist if and only if o = 0, 1, 3, 4, 7 or 9 (mod 12).

Hence cydic directed triple systems can exist only if u = 1, 4 or 7 (mod 12). In fact, the proof of Lemma 3.4 given in [8] can be easily adapted to show that this necessary condition is also suficient; we present a different proof here.

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101

Lemma 3.5. A cyclic DTS(u) exists for u = 1, 4 (mod 12). Proof. A Skolem sequence of order k is a set of pairs {(ai,bi)1 1 s i d k} such that bj - ai = i for 1 =s i s k, and UZ, {ai,bi} = {1,2, . . . , 2 k } . Skolem sequences exist whenever k ‘ 0 , 1 (mod 4) [19]. Let {G,bi} be a Skolem sequence of order k, and consider this set of directed triples: {(;,

j + i, ;+ k + bi)1 0 s ; =s3k, 1 s i s k, arithmetic mod 3k + 1).

It can easily be verified that this is a DTS(3k + 1); since k = 0, 1 (mod 4), the lemma follows. We make the following conclusion.

Theorem 3.6. A cyclic DTS(u) exists if and only if u = 1, 4 or 7 (mod 12). Proof. Necessity follows from Lemmas 3.1 and 3.4. Sufficiency follows from the existence of cyclic STS and Lemma 3.5.

Theorem 3.6 leaves only one case unresolved in the spectrum of regular DTS, that being u = 10 (mod 12). We settle this case by modifying a proof of Bennett and Mendelsohn [2].

Lemma 3.7. A regular DTS(u) exists for u = 4 (mod 6). Proof. Let u = 6 m + 4 ; we construct a DTS(u) with element set {w, x, XI,. . . ,XZ,,,, yo, y 1 , . . . ,YZ,, 20, 2 1 , . . . , z2,,,}. The DTS has the following blocks: (1) (a,xi, yi), (yi, zit x i ) , (xi, m, zi), (zi,yi, m), 0 s i s 2m, (2) (yi-,, xi, yi+,), 0 s i 6 2m, 1 s r s m, (3) (zi+,xi, zi-,), 0 d i =s2m, 1 s r s m, (4) (xi+,, yi, xi-,), 0 Q i S 2m, 1=sr S m, ( 5 ) (zi-,,y , zi+,), 0 s i C 2m, 1 S r d m, (6) (xi-,, zi,xi+,),0 s i d 2m, 1 d r s m, (7) (yi+,,zi, yi-,), 0 s i =S 2m, 1s r G m,

where all subscripts are reduced mod(2m is a regular DTS(6m + 4).

+ 1). It is easy to verify that the result

From Lemmas 3.1 and 3.7 and Theorem 3.6 we have the following theorem. Theorem 3.8. A regular DTS(u) exists if and only

if u = 1 (mod 3).

I02

M.J. Colboum C.J. Colbown

A corollary of this theorem is that refinement of a DTS(v) may fail completely whenever u -= 1 (mod 3).

4. Concluding remarks

Refinement methods, although popular and effective in investigations of graph isomorphism, have not proven appropriate in previous research on design isomorphism problems. For directed triple systems, however, we have found that refinement techniques are well suited to deciding isomorphism. We suggest that this is not an isolated feature of DTS, but rather applies to any graph designs [3, 121 in which the basic graph is asymmetric or, at least, non-transitive; hence, another candidate for refinement techniques would be handcuffed designs [ 12, 141. At present, refinement appears to be inappropriate for graph designs in which the basic graph is transitive, such as STS or Mendelsohn triple systems [15]. We remark in conclusion that one interesting extension of this work on regular DTS would be t o consider other design invariants, and determine the spectrum of designs for which the invariant yields no information regarding isomorphism. Department of Computational Science University of Saskatchewan Saskatoon, Canada

References [ 11 1.. Babai and L. Kucera. Canonical labelling of graphs in linear average time, Roc. 20th Symp. Foundations Comp. Sci. (1979) 39-46. 121 F.E. Bennett and N.S. Mendelsohn. O n pure cyclic triple systems and semi-symmetric quasigroup, A n Combin. 5 (1978) 13-22. [3I J.C. Bermond and D. Sotfeau, Graph decompositions and G-designs, Proc.5th British Combin. Conf. (1975) 53-72. [41 C.J. Colboum, Refinement techniques for graph isomorphism. Proc. 10th S.E. Conf. Combin.. Graph Theory, Computing (1Y79) 281-288. [ 5 ] C.J. Colboum and M.J. Colbourn, Combinatorial isomorphism problems involving l-factorizations. A n Combin. 9 (1980) 191-200. (61 M.J. Colbourn, A n analysis technique for Steiner triple systems, Proc. 10th S.E. Conf. Combin., Graph Theory, Computing (lY79) 289-303. 17) M.J. Colbourn and C.J. Colbourn, Concerning the complexity of deciding isomorphism of block designs, Discrete Appl. Math. 3 (1981) 155-162. 18) M.J. Cidbourn and C.J. Colbourn, Cyclic block designs with block size 3. Europ. J. Combin. 2 (1981) 21-26. 19) M.J. ColbournandR.A.Mathon.Oncyc1icSteiner 2-designs. Ann. Disc. Math. 7(1980)215-253.

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[lo] D.G. Corneil, Graph isomorphism, Ph.D. thesis, University of Toronto, 1968. [ l l ] H. Hanani, The existence and construction of balanced incomplete block designs, Ann. Math. Statist. 32 (1%1) 361-386. [12] P. Hell and A. Rosa, Graph decompositions, handcuffed prisoners, and balanced P-designs, Discrete Math. 2 (1972) 229-252. [13] S.H.Y. Hung and N.S. Mendelsohn, Directed triple systems, J. Combin. Theory Ser. A 14 (1973) 310-318. [14] S.H.Y. Hung and N.S. Mendelsohn, Handcuffed designs, Aequationes Math. 11 (1974) 256-266. [15] N.S. Mendelsohn, A natural generalization of Steiner triple systems, in: Atkin and Birch, eds., Computers in Number Theory (Academic Press, New York, 1971) pp. 323-338. [I61 G.L. Miller, On the ntqsisomorphism technique, Proc. 10th ACM Symp. Theory Comput. (1978) 51-58. [17] R. Peltesohn, Eine h u n g der beiden Heffterschen Differenzenprobleme, Compositio Math. 6 (1939) 251-257. [ 181 R.C. Read and D.G. Corneil, The graph isomorphism disease, J. Graph Theory 1(1977) 339-363. [19] Th. Skolem, O n certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957) 57-68.

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Annals of Discrete Mathematics 15 (1982) 105-109 @ North-Holland Publishing Company

ON THE PRODUCT OF ALL ELEMENTS IN A FINITE GROUP

J6zsef DENES and Pkter HERMANN Dedicated to N.S.Mendekohn for his 65th birthday L. Fuchs proposed the following problem. Given a finite group G of order n, let p(G) denote the set of all elements of the type gl x . . . x g. where {gl,. . .,9). = G. Is it true that p(G) coincides with a c a t modulo the derived group of G ? me authors have solved this problem in the affirmative.

The following problem was due to L. Fuchs: Given a finite group G of order n, let p ( G ) denote the set of all elements of the type gl x * . . x g., where {gl, . . . , g,,} = G. Clearly p ( G )C G'g for a suitable g; does also p ( G )= G'g (i.e., Ip(G)I = IG'l) hold ? In [3, 6; 5, Problem 1.4, p. 4861 one can find the conjecture that the answer is affirmative. For solvable groups it was proved in [7]; some special types of the nonsolvable case were investigated in [I, 21. Using the methods and the main result of [7] we will prove the conjecture for any finite group. (For a consequence of this fact see also [4].) At last we mention a problem also related to our result: Prove that for any finite group G the existence of a complete mapping is equivalent to 1 E p ( G ) . (It is proved for solvable groups. For the definitions and references see [ 5 ] . ) Lemma 1. Let G = (A,x I , . . . , x.) where A 4G and A is Abelian, then

[ A , GI = {[a',X I ] x

- - x [a,,x,,] 1 ai E A } . *

For any subset X = {xi, . . . , xk} C G we will write p ( X )= {xi,X . . . . , i k ) = (1, 2, . . . k}}.

{il,

*

X

xik I

9

Lemma 2. Assume that A
J. Dknes. P. Hetmann

106

For the proofs of the lemmas above see [7, Lemmas 2.2., 2.3. and 2.51.

Lemma 4. Assume N 4G, N' IG'I.

=

N, p ( N ) = N, Ip(G/N)/= IG'/NI, then Ip(G)I =

S be a set of coset representatives of N, G\(N U S ) = { g l , .. . , gn} and go = gl X . * * x g,. Assume that p ( S ) = G'g (mod N), then for every x E G'g, x = y . z where z E p ( S ) and y E N. y E p ( N ) by assumption, so x . go E p(G), hence p ( G ) = G'ggo.

prod. Let

Lemma 5. Assume that G = ( K . K ) with a suitable class K of all conjugates of an involution a. Let C = CG(a), then there exists a set (x", . . . ,x,,} of left representatives of C such that xo = 1 and x;' # xi for all i, j a 1. Proof. Let n E S and S be a Sylow 2-subgroup of G. G = G' by the condition of the lemma, hence IS1 2 4; so we may choose a subgroup T of S such that a E T and IT1 = 4; then T =sS f l C. Suppose that b fi? C i s an involution. Assume that T is Kleinian; clearly ICG(b)n TI s 2 and b . x cannot be an involution whenever x E T\CG(b). If T = (f) is cyclic, then

U ( b * f ) = ' e o(b.f-')=2

@

b-f.b=f-l

+b€C

where cr stands for order. hence b .f and b * f-' are not involutions. Thus in both cases and for all coset t,C 9 C there exist at least two elements y,, z, E r,C such that yf # 1 # zf. Let f denote the graph with the vertexset V = {. . . , y,, . . .} U {. . . , z,, . . .} and the set of edges {. . . . , (z,, y , ) , . . . , (u, v ) / u-I = v}. Since every vertex of f has at most 2 neighbours, V possesses a subset R of nonadjacent vertices andof cardinality fl V(. By thedefinition of r,R U (1) is the desired set of representatives of C. Proposition6. Let a be an involution in a group G and K the set of all conjugates of

a. Assume fhar G = ( K K ), then p ( G ) = G. Proof. Let L = { b l ,. . . , b,} be t h e set of all involutions in G\K, b = bl x

and

b.

= { a . b,

..

*

X

b,

if IK( is even, if IKI is odd.

For any x E G, x = a' X . . . X az,, with a, E K. Choosing n t o be minimal all the a,'s are distinct. In fact. a, = a,, i < j would imply x = al x . . . x a,-l(aia,+lal) X . ..X (ala,-,ar)a,+l x * . x a2,, contradicting t o the minimality of n. I f c E K\{al,. . . , a*,,},we can find an element x such that x-' ' c - x = a. Let

O n the product of 411 elements in

4

finite group

107

Corollary 7 . Let G be a nonabelian simple group, then p(G)= G. Proof. G contains involutions by the Feit-Thompson theorem and for any class K (of conjugates) of involutions ( K * K) is a normal subgroup different from 1.

Theorem 8. (p(G)(= IG'I for every finite group G. Proof. By induction on the order of G. It can be assumed by [7] and Corollary 7 that G is nonsolvable and nonsimple. Suppose that 1 # N 4 G and N is nonsolvable, then 1# (N"))'= N") 4G for a certain integer r (N") denotes the rth term of the derived series). Then all the conditions of Lemma 4 are fulfilled by the inductional hypothesis; so we may assume that all proper normal subgroups of G are solvable. Let N be a maximal normal subgroup of G, then N is the greatest proper normal subgroup. In fact, if M a + G, M $ N, then MIA4 n N = MN/N = G/N, so M would be nonsolvable. By this uniqueness property of N we may assume by Proposition 6 that all involutions of G are in N. As in [7] we define for any normal Abelian subgroup A, H A = (x E G I x 2 fZA). Suppose that A = [A,HA], 1 # A Q G for a suitable A. Let AHA= (A,xl,. . . , x,) where x: $ZA and m is minimal. By the minimality of m there exists a set S of coset representatives of A such that {xl, x;', . . . , x,, x i ' } C S. By our inductional hypothesis p ( S ) = G'y (mod A). Let x E G'y, then x = a z where z E p ( S ) and

-

by Lemma 1. Lemma 2 gives [ai,x i ] E p(ASxiU A*x;'), so

n€p(

u A ' x ~ U A ' X ; ~and )

i<m

xEp

1.Denes. P. Hermann

108

Let {g,, . , . , g,,} = G\(S U

u A " x , U A*x;')

and b = g, x . . . x g, then x . b E p(G),consequently G'yb = p ( G ) . So we may assume that [A.H A ]< A for all 1 # A Q G and A Abelian. G/AHA is elementary Abelian; so AHA is a nonsolvable normal subgroup in G. hence AHA = G. Now G/HA= A/A n HA implies the nonsolvability of H A , thus H A = G. SO we have ( 1 ) [A, GI < A for all proper normal Abelian subgroup A. It is well known that G' commutes with Z2(G)where Z2(G)is the second center of G. As G = G', (2) ZZ(G) = Z ( G ) . One can see that from (1) and (2) immediately follows: (3) Every normal Abelian subgroup of G lies in Z ( G ) . If x. y E G\Z(G) such that 1 f [ x , y ] = c E Z ( G ) , then [ x " , y ] = c" for all integers n. Replace x by a suitable power of x if necessary, and assume that [ x , y] = c is of prime order. By induction hypothesis, p ( G / ( c ) )= G/(c).Use Lemma 3 to get p ( G ) = G. Thus we can assume that for all x, y E G, [x, y ] E Z ( G ) implies [ x , y ] = 1. It follows, in particular, that N is Abelian and hence N d Z ( G ) .Since every involution of G is contained in N we have (4) Every involution of G lies in Z ( G ) . Let S be a Sylow 2-subgroup of G. Suppose that S is Abelian, then N ( S ) > C(S). In fact, N ( S ) = C(S) would imply by the theorem of Burnside the existence of a normal subgroup M of odd order and of index 2" which is impossible. Thus we can find an element a E N(S)\C(S) of odd prime power order. Let H be a nonabelian subgroup of (S, a ) all of whose proper subgroups are Abelian. H is clearly the extension of its normal elementary Abelian Sylow 2-subgroup T by a cyclic group of odd prime power order. This leads by (4) to T s Z ( G ) and H Abelian, contrary to the choice of H. So S is nonabelian; let R be a nonabelian subgroup of S with all proper subgroups Abelian. By the elementary properties of such groups R' = (c) is of order 2, so Ip(G)(= (G'lby (4), Lemma 3 and the inductional hypothesis.

Acknowledgement The authors express their thanks to P.P. PAlfy for the proof of Lemma 5. Inst. for Coordination of Computer Techniques Budapest. Hungary

Depr. of Algebra and Number Theory Elitvijs Lorhnd University Budapest, Hungary

On the product of all elements in a finite group

109

References [l] J. Dtnes, On some properties of commutator subgroups, Ann. Univ. Sci. Budapest Eotviis Sect. Math. 7 (1%) 123-127. [2] J. Dtnes, On commutator subgroups of finite groups, Comment. Math. Univ. St. Pauli 15 (1%7) 61-65. [3] J. DCnes, On some properties of commutator subgroups, Lecture given at the Int. Congr. Math., Moscow, 1966, p. 86. [4] J. Dines and K.H.Kim,On a problem of P. Erdijs and E.G. Straw, Studia Sci. Math. Hung., to appear. (51 J. JXnes and A.D. Keedwell, Latin squares and their applications, Akadimiai Kiad6, Budapest, 1974. [6] J. Dtnes, Research problems, Europ. J. Combin., to appear. [A A.R. Rhemtulla, On a problem of L. Fuchs, Studia Sci. Math. Hung. 4 (1%9) 195-200.

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Annals of Discrete Mathematics 15 (1982) 111-127 @ North-Holland Publishing Company

ENUMERATION OF SYMMETRIC DESIGNS (25,9,3) R.H.F. DENNISTON Dedicated 10 N.S. Mendelsohn on the occasion of his 65th birthday

There are just 78 symmetric designs for which o = 25, k = 9 and A = 3. Three of them have the property of being self-dual but not self-polar.

1. Introduction

When we are told that balanced incomplete block designs exist which have a specified set of parameters, we expect to hear that one of those designs has a fairly large automorphism group. It was even conjectured at one time [4]that there is some transitive group in any such case. Counterexamples t o that conjecture are in fact known; but, when we look at one of the well-known catalogues, we see how usual it is for a design with given parameters to be constructed by allowing a group to act on one or two or three basic blocks. There is, however, one exception that catches our eye in any such catalogue: a symmetric design is known t o exist for which 1) = 25, k = 9 and A = 3, but, short of listing the points of all its blocks, there appears to be no way of specifying it. The design so described, which was found by Bhattacharya [3], is not in fact so shapeless as we might suppose. It is self-dual, with an automorphism group of order 24, under which its points fall into five orbits (1 + 2 + 4 + 6 + 12). But, even so, this is far enough away from the usual situation to raise the question whether a design might exist with these parameters but with a larger group. To establish that n o s&h design exists, we have to exhibit a maximal set of (25,9,3) designs no two of which are isomorphic; and it turns out that there are 78 designs in such a set, and that the automorphism group has in no case a greater order than 24, nor fewer than five orbits on points. Some facts that come up in the enumeration are rather interesting: in particular, three of the 78 designs have the property (for which I know of no other example) of being self-dual but definitely not self-polar. Shaver [lo] has established that there are at least 11 different (25,9,3) designs, but his method is not relevant to the problem of a complete enumeration. This has, however, been made feasible by recent work on another problem. ~

.'

111

R.H.F. Denniston

112.

2. Internal structures

Dembowski [ 5 , p. 31 defined the infernal sfmchcre of a design, with respect to a block 6, as the tactical configuration formed on the points of 6 by the other blocks-this seems a better name than ‘derived design’, which has been used both for this concept and for that of a contraction. In our case, the parameters of such a configuration (which may of course have repeated blocks) are given by u = 9, b = 24, k = 3, r = 8, A = 2. The problem of enumerating configurations with these parameters, after partial solutions by various authors, was almost completely solved by Morgan [8]; she specifies 35 types and explains how to discriminate between them. Mathon and Rosa [7] complete the solution by putting in a thirty-sixth type (which, as it happens, is one of six that never occur as internal structures), and by finding the orders of all the automorphism groups. Isomorphic designs will, of course, have isomorphic internal structures with respect to corresponding blocks. So we have a powerful set of invariants for a (25,9,3) design, if we classify (among the 36 types) its 25 internal structures, and also the 25 of the dual design. To see how easy this makes everything, let us consider the incidence matrix given in Table 1 (for a design isomorphic with the one numbered 27 in Section 6). I am thinking of each column as a block, denoted by a lower-case letter, and having attached t o it the type number of an internal structure: likewise, for each point there is the type number of an internal structure in the dual design. Now, in any anti-automorphism, the point T will have t o correspond t o some block of type 9, namely f or u. If we choose t o have T corresponding to f, we see that u, the block of type 16 through T, can only correspond to W ; and so o n for the other blocks through T, which are all of different types. Then again, the points of u (all fheir types being different) have determinate blocks through W as their images. So we quickly find that T goes to r in a unique antiautomorphism, namely the following permutation of the points and blocks:

( L I M m ) ( N n O o ) ( ~ ) ( R r S s ) ( T I L l uVu ) (W w )(X x Y y ). Likewise, or by group theory, we find that the inverse of this permutation is the unique anti-automorphism taking T to u. And so these two, neither of them being involutory, are all the anti-automorphisms that can act on this design. We haue here an example of a design which is self-dual, but which has no polarity. In fact, the designs numbered 2 and 3 (each differing only slightly from number

Enumeration of symmetric designs (25,9.3)

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R.H.F. Dennislon

1, which is Bhattacharya’s self-polar design) are further examples, although they have respectively 24 and 6 anti-automorphisms. By this method I can easily find whether a given (25,9,3) design is self-dual, exactly what automorphisms it has, and whether it is isomorphic with some design arrived at in a different way. It turns out that a design is characterized by the numbers of its internal structures that belong to the various types; the nearest approach to an exception is that the designs numbered 13 and 21 disagree in the types of one block and two points.

3. Computer searches

The structure types were also needed when I verified that my enumeration of designs was complete. To see how this was possible, let us fix our attention o n a tactical configuration of a definite type, having d pairs of coincident blocks and an autornorphism group of order e (these numbers are available in [7]). And let us suppose we are trying to construct a design having ‘at infinity’, so to speak, a block for which the fixed configuration serves as internal structure. This involves choosing, out of the set of 24 finite blocks, sixteen 9-subsets. each of which is to specify what blocks go through some finite point. Any of these subsets must have just three blocks in common with any other, and also with any of the nine corresponding subsets which already describe t h e geometry at infinity. Suppose we have found that this choice can be made in just c ways. In making this count, we disregard the order in which the 16 subsets are taken; but we do have to pretend we can distinguish between two blocks, even when they go through the same three points at infinity. Since in fact two such blocks could very well be interchanged, we must say that the structure at infinity is invariant under 2de permutations of that set of 24 blocks from which we have been choosing. Suppose, o n the other hand, that we have solved the problem of enumerating (25,9,3) designs-that we can see one representative of each of their transitivity classes under isomorphism. Let the ith representative (with an automorphism group of order gi) include fi blocks for which the internal structures are of our fixed type. Then a familiar counting argument establishes the equation

Accordingly, when I had constructed a set of 78 different designs, and wondered whether it was complete, I had to see if this equation would come

Enumeration of symmetric desigm (25,9,3)

115

out right for each type of configuration. In every case but one, I could easily determine the value of c by making searches on the Cyber 73 computer at the University of Leicester. If the arrays of my program would not hold all the possibilities arising in some case, I subdivided it into manageable cases, usually by requiring one or two finite points (as well as the nine points at infinity) to lie on specified sets of blocks.

4. The repeated a f i e plane

Such a method, however, would have been very awkward to apply to Morgan’s type 1 (the configuration which we get from an affine plane of order 3 by counting every line as two coincident blocks). Instead of putting this case into the machine, I settled it by an argument of which the length may perhaps be excused by the interest of the conclusion. Bhattacharya’s strategy was: first to construct, on 16 points, a design capable of serving as external structure (residual design) with respect t o a block where the internal structure was to be of type 1; and secondly t o fit the two structures together into a (25,9,3) design. I shall now show that, although he could have chosen from eight different solutions t o his second problem, the solution to the first one was uniquely determined. What, in fact, must the ‘finite’ structure look like, if the structure ‘at infinity’ is to be of type 1 ? For each line of the affine plane there will be two finite blocks, a ‘pair of parallels’, say, meeting in three points at infinity and in no finite point. For each class of three disjoint lines (it would be inconsistent if we called it a ‘parallel class’) there will be a hexad of finite blocks, made up of three pairs of parallels. Two non-parallel blocks, if they are in the same hexad, will have three common finite points, since they do not meet at infinity; two blocks from different hexads have in common one point at infinity and two finite points. Let us denote the hexads by c5, &6, c7, &g (I must keep the suffixes I to 4 for another purpose). Now consider the nine blocks through a given finite point P. No two of them are parallel, so they meet the plane at infinity in nine different lines. Each point at infinity, being joined to P by three blocks, is on three of these lines; so the remaining three lines at infinity are such that each point at infinity is on just one of them. This means that, given any finite point P, we can find one of our four hexads that has none of its blocks going through P. Therefore the 16 finite points fall into four sets T ~ 76, , T ~ 7,8 ; a point of T~is on no block of ci,but on one block of each pair of parallels in E~ (i # j). Counting in two ways the incidences of blocks of tz5 with finite points, we see that

R.H.E Dcnniston

116

1~~

U T~ U ~~1 = 12. So the sets

7;

are of four points each, and we may call them

‘tetrads’. Now suppose if possible that two points P, 0 of r5 lie on two blocks a, b of E ~ Let . R be the third (finite) point common to a and b; let c be the third block joining P and Q, and let a‘, b’, c’ be parallel to a, b, c. Where are the nine blocks through R ? They could include c‘, but certainly not a’ nor b‘. The remaining twelve blocks of &6 Ue7 U &8 each go either through P or through Q, and R is joined to each of P and 0 by one more block besides a and 6 ; so R is on two of the twelve and not on c (or, perhaps, on c and on none of the twelve). This gives us at most five blocks of &6UE7U&8going through R. But this is not enough, since R is on (at most) one block from each of the three pairs of parallels in E ~ So . the supposed incidences are impossible. Therefore, when we consider the 12 incidences of points of ri with blocks of E, ( i # j ) , we find that there is only one possibility: any two of the four points are joined by just one of the six blocks (with the parallel block joining the other two). We may describe these points and blocks as ‘the vertices and sides of the quadrangle aii’.Two non-parallel sides of a quadrangle will have a vertex in common. So, if the parallel pairs of blocks in &8 are, say, f’}, {g, g’}, {h, h’}, there is one vertex of assthat lies on f and g-and also on either h or h‘. In the case where @, g, h} is the triad of sides through this vertex, the triads through the other three vertices are determined as g’, h’}, g, h’}, g’, h}: and the only other possibility is that the triads are g’,h’}, g, h}, cf, g’, h}, g, h’}. that have the same And so, when we look at two quadrangles like ass and six blocks as sides, there are two contrasting possibilities-the quadrangles may be ‘unlike’ (one having a vertex o n f , g, h, and the other on f’, g’, h’) or ‘like’. Furthermore, to each point (A. say) of r5 there corresponds a definite point ( X , , say) of r6,such that, of the three blocks of E~ that go through A, either all or none go through XI. But A and XI must be joined by three blocks (which belong neither to E~ nor to & 6 ) ; so, of the three blocks of E~ through A, none or all go through XI. Again, if A corresponds to a point Yl of r7,we see (looking ) XI and Yl correspond in the same at their incidences with blocks of E ~ that manner. In this way we find that the 16 finite points can be thought of as a 4 x 4 array, the rows being the tetrads, and any two points in the same column corresponding as described in the last paragraph:

u,

u, v,

v, v, v,

u,

Enumeration of symmetric designs (25,9,3)

117

I must explain that I feel obliged to use Bhattacharya’s names for these points, since what I am in the middle of proving is that he has solved his first problem in the only possible way. a@} and { c T ~ ~g 6,7 } of We have seen in passing that, of the pairs {ass, quadrangles, one must be a like and the other an unlike pair. Also, of the three quadrangles us,a@,am that use the blocks of as sides, either all are alike, or two are unlike the third but like each other. Now any like pair of quadrangles Y l ,Z1with the three blocks that they need to provides two of the points A, XI, join them; so there must altogether be just six pairs of like quadrangles. This means that just one of the hexads has the property that its three quadrangles are all alike: we may suppose (changing the notation if necessary) that this is the hexad c5. (So T~ is different from the other three hexads, as Bhattacharya has emphasized by naming its points from the other end of the alphabet.) Making another change of notation if necessary, we may suppose it is {ass, aa}that is a like pair and a67} that is unlike. This determines that a 7 8 , aesand a 6 7 are the three quadrangles of which we can say that each is unlike the other two quadrangles using the same blocks as sides. But now we know which finite points are incident with any finite block. If we consider, for instance, the side of the quadrangle as7that joins A and C, we see that in the like quadrangle it will go through the points Z1and Z3 in the same two columns of the array: but, in the unlike quadrangle a 6 7 , it will go, not through the corresponding points X , and X3, but through the other two vertices X, and X4.If, with respect to some block of a (25,9,3) design, the internal structure is to be a repeated afine plane, the only possibility for the external structure is that it is isomorphic with the one specified by Bhaitacharya. We have now to consider how the two structures are fitted together. What is significant for both of them is the separation of the 24 finite blocks into hexads and into pairs of parallels; and then, on the one hand, we have four pairs of parallels incident with each point at infinity. On the other hand, we have , imposed a cyclic order on &6, E ~ E, ~ Besides . distinguished the hexad E ~ and that, we have eight blocks (one pair of parallels from each hexad) which, as sides of quadrangles, join vertices with suffixes 1 , 2 or 3,4; eight that join 1,3 or 2,4; and eight that join 1,4 or 2,3. Perhaps the simpler method will be to start at infinity. We are given, then, an affine plane of order 3, in which one of the four classes of disjoint lines is distinguished, and so is a cyclic order of the other three classes; this restricts us to a subgroup of index 8 in the collineation group of the plane. And we are to partition the lines into three sets, each including one line from each class, in as many different ways as possible (ways, that is, of which no two are equivalent under the subgroup). This is an easy problem to solve by hand-and the number of different

118

R.H.F. Denniston

partitions comes o u t to be eight. On the other hand, we have in Section 6 eight different designs (numbered 1, 2, 5 , 6, 9, 10, 16 and 24), each having for one of its blocks a type-I internal structure. Thus the last sentence of the first paragraph in this section is justified, and so is my refusal to search (as in Section 3) for the number c associated with the configuration of type 1. This is not the only case known in which two non-isomorphic symmetric designs have, with respect to one block of each, isomorphic internal and external structures: the same thing happens with the (71, 15, 3) designs discovered by Haemers [6, p, 691. But the present example is even further from being analogous to the uniqueness of the embedding of an affine in a projective plane. We may also observe that Beker and Piper [2] have generalized the pattern we see here, and so constructed a (possibly infinite) family of symmetric designs.

5. Switching ovals

The method of Section 3 could have been used, in a rather tedious way, for the construction of a set of different designs, as well as for the verification that such a set was complete. But in fact I carried out the construction in a more interesting way. Suppose we have found, in some (25,9,3) design already known, a set {P, Q. R. S} of four points no three of which lie in any block. Then P will be joined to 0.R, S by nine blocks that are all different-and there are only nine blocks through P altogether. So the set is an ‘oval’, as that concept has been generalized by Assmus and van Lint [l]; or it might be more relevant t o call it a ‘maximal arc’. as that concept has been generalized by Morgan [9]. Whatever we call it, the property we need is that any block can only meet the oval in two points or in none at all. Suppose, then, that we take the three blocks through P and Q, and the three through R and S, and change their incidences: make the former three go through R and S, and the latter three through P and Q, their incidences with points outside the oval being unaffected. Then we still have a design. (To prove this, we need only consider a block a which is one of the six and a block b which is not, and make sure that there are still just three points common to these two, as there were before. And in fact, if b goes through no point of the oval. its intersections with Q are quite unchanged: if b meets the oval in two points, it has lost one intersection and gained one.) For instance, the tetrad {A,B,C, D } of Section 4 is an oval. In listing Bhattacharya’s design as number 1 in Section 6 below, I have kept the names

Enumeration of symmetric designs (25, 9.3)

119

A, I?, C, D for these four points, though compactness requires that the symbols for the other points shall be letters without suffixes. And then, by ‘switching’ this oval as just explained, we go from design number 1 to number 2, which was mentioned in Section 2 as being self-dual but not self-polar. It was actually possible, by repeated use of the operations of switching an oval, and of going over from one design to its dual, to generate a set of 78 different designs from Bhattacharya’s one. I used the computer to search for ovals in each successive design, and to do most of the work of classifying internal structures into Morgan’s types. For some time, whenever a newly generated design appeared from its structure types to be isomorphic with an old one, I verified by hand that all the incidences agreed; but then I realized that the work of Section 3 would have to be done eventually, and would show up any false assumption of isomorphism. In the classical case of the Steiner triple systems of order 15, a switching process (analogous to the dual of mine) generates a set of 79 systems from the projective geometry, but there is an eightieth system which cannot be reached in this way. Nothing so dramatic happens with our problem: the equation of Section 3, with i running from 1 to 78 over the generated set of designs, always comes out right. And so the conclusion is, as asserted in Section 1 , that up to isomorphism, there are just 78 symmetric designs (25,9,3).

6. Catalogue of designs

I must now exhibit the solution of the problem by writing out various sets of 25 blocks. I give in each case the order of the automorphism group, and the number of lines (the concept of a line is here self-dual, involving the incidence of three points with three blocks). I also say how many ovals there are, and into which designs the given one can be changed by switching them. Where a design and its dual are non-isomorphic, I list the blocks of one of them, adding in parentheses a serial number for the other and information about its ovals. The other two numbers are, of course, the same for both. Design numbered 1. Group order 24; 28 lines; 16 ovals, switching to designs numbered 2, 3, 4, 6. HuNopruv EFGNOPQRS CDEHKORTW ACFILNR’IX ADGHLOSVX

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ADGIKPQTY

BCGJMOQTX

17. 2; 12; 10, to 5, 22, 25, 26, 27, 31, 33. HlJNOFRJV EFGHKNOQR CDFKORTW ACENPRSTX ADGHLOSVX

HIJQRSWXY EFGTUVWXY CDEHIKMUX BDFHM”STY ADGJMNRUW

KLMNOPWXY ABElLNQUX CDETLNOW BDFJKPRW BCGIKNSVW

ACEHMWVW BDEOWSUW BCGHLPRW

18. 2; 12; 10, to 5, 22, 25, 26, 27, 31, 36. GHIJL.0PW EFGNOPQRS CDFILORTW ACEGHKLRX ADHNOSTVX

HlJQRSWXY EFGTUVWXY CDELMPSUX BDFGHLMSY ADGJhfNRUW

KLMVOPWXY ABRUNQUX CDEnNQW BDFJKPRVX BCGIKNSVW

KLMQRSTUV ABElLpsTw

ACFHMPQVW BDEHKOQUW BCHNPRTUY

EFHlJKMNT ABEMORW ACFJKOSW

ADGIKPQTY

BCGJMOQTX

19 (20). 1; 11; 16. to 7, 8, 11, 13, 37 (7, to 12, 39, 41, 43, 44). HIJNOPTUV ACFHIWOQR ABEOPRSTW EFGLLNRTX ADGHLOSVX

HLIQRSWXY A m ABEHIKMUX BDFHMNSTY ADGJMNRUW

KLMNOPWXY CDENWSUX ABEJINQW BDFJKPRVX BCGIKNSVW

KLMQRSTUV CDEmTW EFGHMPQVW BDFILOQUW BCGHLPRW

ACFIJLMPS

CDEIMORW EFGJKOSW ADGMPQTY BCGJMOQTX

21 (22). 2; 14; 16, to 7, 10, 11, 13, 37 (10, to 9, 14, 17, 18, 42, 45, 46). HLlNOYWV ACFOPRSTW ABEHKNOQR EFGLLNRTX

ADGHLOSVX

HIJQRSW ACiWQUVXY ABELMPSUX BDFHMNSTY ADGJMNRUW

KLMNOPWXY CDEHKTUHSY ABElLTVWY BDFJKPRVX BCGIKNSVW

KL.MQRSTUV CDEJL.NPCS EFGHMWVW BDFILOQUW BCGHL.PRW

ACFWJKLM CDEIMORW EFGJKOSW

ADGIKPQTY

BCGJMOQTX

R.H.F. Denniston

122

23 (24). 12; 22; 16. to 8, 10, 15, 37 (10, to 9, 16, 46). HIJNOPTUV EFGNOQR V Y ABEOPRSTW ACFILNRTX ADGHLOSVX

HIJQRSWXY EFGHKTLWX ABEIMUVXY BDFHMNSTY ADGJMNRUW

KLMNOPWXY CDENPQSUX ABEHJKLNQ BDFJKPRVX BCGIKNSVW

KLMQRSTUV CDEJLTVWY ACFHMPQVW BDFlLOQUW BCGHLPRUY

EFGIJLMPS CDEHIKh4OR ACFJKOSW ADGIKPQW BCGJMOQTX

25 (26). 2; 12; 10. to 9, 14, 17, 18. 45, 46, 47 (10, to 14, 16, 17, 18, 40, 46). GIUMNPTV EFGNOPQRS CDFIL O R TW ACEHKNRTX ADGHLOSVX

KLMQRSWXY EFGTUVWXY CDEIMPSUX BDFHMNSTY ADGJMNRUW

HIJNOPWX Y ABFILNQUX CDEJLN O W BDFJKPRVX BCKNOSUVW

HIJQRSTUV ABEJLPSTW A CFHMPQ V W BDEGHIKQW BCGHLPRW

EFHJKLMOU ABEIMOR VY A CFGIJKSY ADKOPQTUY BCGJMOQTX

27. 2; 12; 10, to 9, 14, 17, 18, 42, 46, 47. KLMNOPTUV EFGHKNOQR CDFILORTW ACENPRSTX ADGHLOSVX

KLMQRSWXY EFGTUVWXY CDEHIKMUX BDFHMNSTY ADGJMNRUW

HIJNOPWXY ABFILNQUX CDEnNQVY BDFJKPR V X BCGIKNSVW

HIJQRSW ABEHJUTW ACFHMPQVW BDEOPQSW BCGHLPRW

EFGIJLMPS ABEIMOR W ACFJKOSW ADGIKPQTY BcG/MOOTX

28 (29). 3; 10; 16. to 11, 13, 15, 37 (4, to 38, 48). HIJNOPTUV A C F N W R VY CDEOPRSTW EFCILNR 'LX ADGHLOSVX

HIJQRSWXY ACFHKTUWX CDEIMWXY BDFMSTY ADGJMNRUW

KLMNOPWXY ABENPQSUX CDEItrKLNQ BDFJKPR V X BCGIKNSVW

KLMQRSTW ABEJLTVWY EFGHMPQVW BDFILOQUW BCGHLPR W

ACFIJLMPS ABEHIKMOR EFGJKOSW

ADGIKPQTY

BCGJMOQTX

30 (31). 1; 5; 7, to 12, 39, 49, 50, 52 (10, to 17, 18, 40, 42, 45, 47, 53, 55). EFGHlJTUV EFGNOPQRS CDFILORTW ACEGHKLRX ADHNOSTVX

HIJQRSWXY GLOPLNUXY CDEIMPSUX BDFGHLMSY ADGJMNRUW

EFKMNTWXY ABFILNQUX CDEJLNQW BDFJKPR V X BCGIKNSVW

KLMQRSTUV ABEJLPSTW ACFHMWVW

BDEHKOOUW BCHNPRTUY

HIJKLMNOP ABELMOR W ACFJKOSW ADGIKPQTY BCGJMOQTX

32 (33). 1; 6; 7, to 12, 41, 43, 44, 49 (10, to 14, 17, 40,42, 45, 47, 53, 55). EFGHlSTuV EFGNOPQRS CDFILORTW ACENRTVXY ADGHLOSVX

HlJQRS W X Y HKNOPTUWX CDEIMPSUX BDFHMNSTY ADGJMNRUW

EFGKLMWXY ABFILNQUX CDEWUNQ BDFJKPRVX BCGIKNSVW

KLMQRSTUV ABEJLPSIW ACFHMFQVW

BDEOQUVWY BCGHLPRW

IJLMNOP W ABEHlKtvfOR ACFJKOSW ADGIKPQTY BCGJMOQTX

34 (35). 1; 5 ; 7, to 12, 41, 43, 44,SO (7, to 12, 41, 43, 44,52). EFGHIJTW EFGNOPQRS CDFILORTW ACEHKNRTX ADGHLOSVX

HIJQRSWXY NOpruvwXY CDEIMPSUX BDEMMNSTY ADGJMNRW

EFGKLMWXY ABFIJLNPX CDULNQW BDFKQRUVX BCGIKNSVW

JKLMPRSTV ABELQSTUW ACFHMPQVW

BDEWKOPW BCGHL.PR W

HIKLMNOQU ABEIMOR W ACFJKOSW ADGIKPQTY BCGJMOQTX

123

Enumeration of symmetric designs (2?5,9,3)

36. 1; 6; 10, to 14, 18, 40, 42, 45, 47, 53, 55. FGHIOTWX EFGNOPQRS CDEHKORTW ACGJMNRZY ADEGHJLSV

HIJQRSWXY EJNFlVVWY CDEIMPSUX BDFHMNSTY ADFILNRLIW

EFGKLMWXY ABEHKNQUX CDLNOQVXY BDFJKPRVX BCGIKNSVW

KLMQRSTW ABLOPSTWX ACFHMWVW BDGJMOQUW BCGHLPRW

HIJFLMVOP ABEIMOR W ACFJKOSW ADGIKPQTY BCEFIJLQT

37 (38). 3; 10; 16, to 19, 21, 23, 28 (4, to 29, 56). HIJNOPTIJV' ACFNOQR W ABEOPRSTW EFGILNRZY ADGHLOSVX

HIJQRSWXY AG7WKTUU.X ABEIMWXY BDFHMNSTY ADGJMNRLTW

KLMNOPWXY CDENPQSUX ABEHJKLNQ BDFJKPRVX BCGIKNSVW

KLMQRSTW CDEJLTVWY EFGHMPQVW BDFILOQUW BCGHLPRW

ACFIJLMPS CDEHIKMOR EFGJKOSW ADGIKPQTY BCGJMOQTX

39 (40). 1; 5; 7, to 20, 30, 58, 60,62 (10, to 26, 31, 33, 36, 57, 64, 65). EFGKLMTUV EFGNOPQRS CDFILORTW ACEHKNRlX ADGHLOSVX

KLMQRSWXY GINPrVWXY CDEIMPSUX BDFHMNSTY ADGJMNRW

EFHJOWXY ABF7LNQUX CDEJLNQW BDFJKPRKX BCKNOSUVW

HIJQRSTUV ABEJLPSTW ACFhIMWVW BDEGHIKQW BCGHLPRW

HIJKLMNOP ABEIMORW ACFGIJKSY ADKOPQTUY BCGJMOQlX

41 (42). 1; 6; 7, to 20, 32, 34, 35, 58 (10, to 22, 27, 31, 33, 36, 57, 61, 67). EFGKL.MTUV EFGNOPQRS CDFILORTW ACENRTVXY ADGHLOSVX

KLMQRSWXY HKNOPTUWX CDEIMPSUX BDFHMNSTY ADGJMNRUW

EFGHlJwxY ABFILNQlLX CDEHJKLNQ BDFJKPRVX BCGIKNSVW

HIJQRSTW ABWLPSTW ACF'HMPQVW BDEOQLNWY BCGHLPRW

IJLMNOPW ABEHIKMOR ACFJKOSW ADGIKPQTy BCGJMOQITX

HIJQRSTUV ABWLPSTW ACFHMFQVW BDEHKOQUW BCGHLPRW

HlKLMOPSX ABEIMOR V Y ACFJKOSW ADGIKPQTY BCGJMOQTX

HIJMPQS W ABWLPSTW ACFHQRTVW BDEHKOQUW BCGHLPR W

HIJKLNORT ABEIMOR V Y ACFJKOSW ADGIKPQTY BCGJMOQTX

43. 1; 5; 7, to 20, 32, 34, 35, 60. EFGKLMTUV EFGNOPQRS CDFILORTW ACEHKNRIIX ADGHJLNOV

JKLMNQRWY NOPTUVWXY CDEIJMNPU BDEHMNSTY ADGMRSUWX

EFGHIJWXY ABFILNQUX CDELQSVXY BDFJKPR VX BCGIKNSVW

44. 1; 5; 7, to 20, 32, 34, 35, 62. EFGKLMTUV EFGNOPQRS CDFlLMOPW ACEHKMNPX ADGHLOSVX

KLMORSWXY NOPTUVWXY CDEIRSTUX BDFHMNSTY ADGJMNRUW

EFGHUWXY ABFILNQUX CDEJLNQW BDFJKPRVX BCGIKNSVW

45. 1; 6; 10, to 22, 25, 31, 33, 36, 57, 61, 67. E F G K W EFGNOPQRS CDKOpRTVW ACFILNRZY ADGHLOSVX

KLMQRS WXY EHNOTUWXY CDEIMPSUX BDGIKNSTY ADGJMNRUW

FGIJPVWXY HIJQRSTUV ABKNPQUVX ABWLPSTW CDEJLNQW ACEGHIKQ W BDEFHJKRX BDFILOQUW BCFHMNSVW BCGHLPRW

HIJKLMNOP ABEIMORW ACFJKOSW ADF%UPQTY BCGJMOQTX

R.H. F. Denniston

124

46. 2; 12; 10, to 22, 24, 25, 26, 27, 57. EFGKLMTUV EFGNOPQRS CDEHKORTW ACFILNRTX ADGHLOSVX

KLMQRSWXY JLNOPTUVWY CDEIJLMPS BDGIKNSTY ADGJMNRW

EFGHIJWXY ABEHJKLNQ CDENQUVXY BDFJKPR V X BCFHMNSVW

HIJQRSTUV ABEPSTUWX ACGIKPQVW BDFILOOUW BCGHLPRW

HIKMNOPUX ABEIMORW ACFJKOSW ADFMMPQTY BCGJMOQTX

47 (48). 1; 5 ; 10, to 25, 27, 31, 33, 36, 57, 61, 67 (4, to 29, 56, 59). GIKLMNPTV EFGHKNOOR CDFILORTW ACENPRSTX ADGHLOSVX

KLMQRSWXY EFGTUVWXY CDEMKMUX BDFMNSTY A D G J M N RW

HIJNOPWXY ABFILNQUX CDEJLNQW BDFJKPR V X BCKNOSUVW

HlJQRSTUV ABEIUKLTW ACEHMPQVW BDEGIWSW BCGHLPRUY

EFJLMOPSU ABEIMORW ACFGIJKSY ADKOPQTUY BCGJMOQTX

KLMQRSTW ABEJLPSTW ACFHMPQVW BDEOOUVWY BCHNPRTUY

lJLMNOPW ABEHIKMOR ACFJKOSW ADGIKPQTY BCGJM0QT.X

49. I ; 2: 7. to 30, 32, 58, 60,62. EFGHISTUV EFGNOPQRS CDFILORTW ACEGLR V X Y ADHNOSTVX

HIJQRSWXY GHKLOPUWX CDEIMPSUX BDFGHLMSY ADGJMNRW

EFKMNTWXY ABFILNQUX CDEHJKLNQ BDFJKPR V X BCGIKNSVW

50 (51). 1; I ; 7, to 30. 34. 58, 60,62 (4, to 63, 68,69). EFGHISTUV EFGNOPQRS CDFILORTW ACEGHKLRX ADHNOSTVX

HIJQRSWXY GLOPUVWXY CDEIMPSUX BDFGHLMSY ADGJMNRlJW

EFKMNTWXY ABFlJLNPX CDEJLNQW BDFKQRUVX BCGIKNSVW

JWPRSTV ABELQSTUW ACFHMPQVW BDEHJKOPW BCHNPRTUY

HIKLMNOQU ABEIMORW ACFJKOSUY ADGIKPQTY BCGJMOQTX

KLMQRSTUV ABEJLMOPT ACFHMPQVW BDEHKOQUW RCHh'PRTUY

HIJKLNPSW ABEIRSVWY ACFJKOSUY ADGIKPQTY BCGJQSTWX

52. 1; 1; 7, to 30, 35, 58. 60,62. EFGHIITW EFGNOPQRS CDFILORTW ACEGHKLRX ADHNOSTVX

HIJMOQRXY GLOPUVWXY CDEIMPSUX BDFGHLMSY ADGJMNRLJW

EFKMhTWXY ABFILNQUX CDHLNQW BDFJKPRVX BCGIKMNOV

53 (54). 1; 1; 10, to 31, 33, 36, 61. 64, 65, 67 (4, to 63, 66, 68). HUNOFTUV ACFHKNOQR CDEOPRSTW EFGILNRTX A DGHLOSVX

HIJQRSWXY ABFGHPTWY DEGHIKMPQ DFMNQSTVY ADGJMNRLW

KLMNOPWXY ABENPQSUX BCDEHJLNY BDFJKPRVX BCGIKNSVW

CEFMMUVWX BDFILOOW CGLPQRUW

ACFIJLMPS ABEIMORVY EFGJKOSW ACDIKTUXY BCGJMOQITX

KLMQRSTUV AEWLQTWY CEFHMUVWX BDFILOQW BCGHLPRW

ACFIJLMPS ABEIMORW EFGJKOSUY ACDIKTUXY BCGJMOQTX

BHKLMRSTU AEJKLQTVW

55. 1; 1: 10, to 31. 33. 36, 61, 64,65, 67. HIJNOPTW

A CFHKNOQR

CDEOPRSTW EFGKNRTX ADGHLOSVX

BHIJKRSWX ABFGKPTVW DEGHIKMPQ BDFHMNSTY ADGJMNRW

KLMNOPWXY ABENPQSUX BCDEJKLNV DFJPQR V X Y CGINQS VWY

Enumeration of symmetric designs (25,9,3)

125

56 (57). 1; 5; 4, to 38, 48, 71 (10, to 40,42, 45, 46,47, 72, 74). FGKLNTUVW EFGNOPQRS CDGJMORTW ACEHKNR'IX ADGHLOSVX

KLMQRSWXY EMOPTUVXY CDZh'??SUUK BDEHMNSTY ADEETLMRU

EFGHIJWXY ABGJMNQm C"QW BDFJKPRVX BCEGIKMSV

HUQRSTUV ABETLPSTW ACFHMPQVW BDEHKOQUW BCGHLPRW

HLJIUMNOP ABINORVWY ACE?KOSW ADGIKPQTY BCFLLOQlX

58 (59). 1; 2; 7, to 39, 41, 49, 50, 52 (4, to 48, 71, 75). EFGKLMTUV EFGNOPQRS CDFKORTW ACENRTVXY ADGHLOSVX

KL.MQRSWXY GHIKNPTUX CDEIMPSVX BDEMMNSTY ADGJMNRUW

EFhuOUWXY ABFILNQUX CDEHIKLNQ BDFJKPRVX BCKNOSLIVW

IJLMNOPW ABEHIKMOR ACFGLIKSY ADKOWTUY BCGJMOQTX

HIJQRSTUV ABEILpsrw

ACFHMPQVW BDEGIQVWY BCGHLPRW

60 (61). 1; 1; 7, to 39, 43, 49, 50, 52 (10, to 42, 45, 47, 53, 55, 72, 74). EFGIUMTUV EFGNOPQRS CDFLLORTW ACEHKNRlX ADGWLNOV

JMQRWY GINPTVWXY CDEUMNPU BDFHMNSTY ADGMRSW

EIWJOWXY ABmNQUX CDELQSW BDFJKPRVX BCKNOSUVW

m m o p s x ABEIMORVY ACFGIJKSY ADKOPQTUY BCGJMOQTX

HIJQRSTUV ABEILPSTW ACFHMPQVW BDEGHIKQW BCGHL.PRW

62 (63). 1; 1; 7, to 39, 44,49, 50, 52 (4, to 51, 54, 76). EFGKLMTUV EFGNOPQRS CDFlLMOPW ACEHWPX ADGHLOSVX

IUMQRSWXY GINPTVWXY CDEIRSTUX BDEHMNSTY ADGJMNRUW

EFWOWXY ABITLNQUX CDEnNQW BDFJKPRVX BCKXOSUVW

HUMWSW

ABEJL.PSTW A-QRTVW BDEGHKQW BCGHLPRW

HIJKLNORT ABEMORVY AeGLJKSY ADKOPQTUY BCGMOQTX

KLMNOPWXY BCDE3NPQX AELNQSLJVY DFKPRSWX BCGIKNSVW

BJIUMQRTV DEGmOS EFGHMK?VW BDFLLOQUW BCGHLPRW

A m W S CDEIMOR W CEFXTUWY ADGIKPQTY CGMOQSTUX

64. 3; 1; 10, to 40,53, 55, 74. HuNopnrv ACRiKNOQR ABEOPRSTW EFGENRTX ACDHLTVWX

HIJQRSWXY ABFG3OVXY ABEHIKMUX BDEMMNSTY ADGJMNRUW

65 (66).3; 0; 10, to 40,53, 55, 72 (4, to 54, 77). HIJNopruv ACFHKNOQR ABEOPRSTW EFG1L.NRlX ACDHLTVWX

BHULQRWY ABFGLOUVY ABEHIKMUX BDFHMNSTY ADGMRUW

KLMNOPWXY BCDELNPQU AEYNQSVXY BDFJKPRVX BCGIKNSVW

KLMQRSTUV ACFIJL.MPS DEGHJKLOS CDEIMORW EFGHMPQVW CEFJKTiYWY DFIOQSUWX ADGIKPQTY CGHPRSLIXY BCGJMQX

67 (68). 1; 1; 10, to 42, 45, 47, 53, 55, 72, 74 (4, to 51, 54, 77). HLJNOPTWV ACFHKNOQR ABGJMORTW EFGZNRlX

ADGHLOSVX

HLIQRSWXY A(-'FTUVWXY

ABEHIKMUX BDFHMIVSTY ADEFJPRSU

FJKLMOPXY CMJMNGWX ABElLNQW BDWVPRW BCFGIJKSV

KL.MQRSTUV CDE€DKLTW EFG-VW BDFILOQUW BCGHLPRW

ACILMhPSW CDEMORVY EGKNOSUWY ADGIKPQTY BCEOPQSlX

R.H.F. Denniston

126

69 (70). 3; 0; 4. to 51, 76 (1, to 73). HIJNOPTUV .4CFHKNOQR CDEGHPRST DFLNOSTWX AEGIL.OR L'X

HIJQRSWXY AFGPQTVWY D E I K M o p oW BEFIMNRTY A D G J M N R UW

GHKLMNPXY ABENPQSUX CDFJLNQVY BDFJKPR V X BCGIKNSVW

KLMQRSTUV ABEHJKLTW CEFHMLNWX BDFGHILQU BCLOPR UWY

EFHJOUWX Y ABGJMNQUX CDWLNQW BDFJKPR V X BCEKMOSUV

HIJQRSTUV ABEILPSTW

ACFIJLMPS ABDHMOSVY EFGJKOSUY ACDIKTUXY BCGJMOQTX

71. 1: 2; 4, to 56, 59, 78 FGK L NTIJVW EFGNOPQRS CDGJMORTW ACEHKNRIIX ADGIZLOSVX

K L MQRS WXY EGIMPTVXY CDINPSUWX BDFHMNSTY ADEFILMRLJ

BDEGHIKQW BCGHLPRUY

HIJKLMNOP A B I N O R VWY ACFGIJKSY ADKOPQTW BCFILOQTX

KLMQRSTUV CDULTVWY AEFHJLMPQ BDFILOQW BCGHLPRW

CDFGHIMSV CEIKMOPRW EFGJKOSUY ADGIKPQW BCGJMOQIX

ACFHMWVW

72 (73). 3 ; 0; 10. to 57, 61, 65. 67 (1, to 70). HIJNOPTUV ACFNOQR VY ABDEHORST EFGILNRTX AGL0PSVW.Y

HIJQRSWXY ACFHKTUWX ABEIMUVXY BFMNPSTWY ADGJMNRUW

DHKLMNOXY CDENPQSUX BEGHKNQVW BDFJKPR V X ABCIJKLNS

74 (75). 3; 1 ; 10, to 57, 61, 64,67 (4, to 59, 78). HIJNOPTUV ACFNOQR VY ABDEKOPRT EFGILNRTX ADGHLOSVX

DHIJKQRXY ACFHKTUWX ABEIMUVXY BDIWMNSTY ADGJMNRUW

KLMNOPWXY CDENWSUX BEGHKNQVW BFJPRSVWX ABCIJKLNS

KLMQRSTUV CDEILTVWY AEFHJLMPQ BDFILOQW BCGHLPR W

CDFGIKMPV CEHIMORSW EFGJKOSW AGIFQSTWY BCGJMOQTX

GHKLMOPWY CDEGHFQSU ABDJKPQW BEFGWLRV BCGIKNSVW

KLMQRSTUV DEJKLNOSX EFUNPOVWX BDFILOQUW BCLNPR U X Y

ACFIJLMPS CDEIMOR VY CEFJKTUWY AEGILNQTY BCGJMOQTX

EFHJOUWXY AFKLNQTUX CDWLNQVY BDFJKPR V X BCKNOSUVW

HIJQRSTUV ABEJLPSTW ACFMPQVW DEIWQSWX CLMPRSUXY

HIJKLMNOP AEKMORTVY ACFGIJKSY ABDIOPOW BCGJMOQ7X

76. 3; 0; 4, to 63, 69. HIJNOPTCJV ACFHKNOQR ABEOPRSTW DFGIKPRIX AC'DHLTVWX

HIJQRS W X Y AFGOSUVXY ABEHIKMUX BDFHMNSTY A D G J M N R UW

77. 3 ; 0 ; 4. to 66, 68. BEFGILMUV EFGNOPQRS CDFILORTW .4BCEHINRX ALSHLOSVX

BGHKLQR WY GINPNWXY CDEGHKPTU BDFHMNSTY A U G J M N R UW

Enumeration of symmetric designs (25,9 , 3 )

127

78. 3; 1; 4,to 71, 75. FGKLNTUVW EFGNOPQRS CDGJMORTW ACENRTVXY ADGHLOSVX

KLMQRSWXY EGHIKMPTX CDINPSUWX BDMNSTY ADEFILMRU

EFHJOUWXY ABGJMNQUX CDEWKLNQ BDFJKPR V X BCEKMOSUV

HIJQRSTUV ABEJLPSTW ACFHMPQVW BDEGIQVWY BCGHLPR W

IJLMNOPW ABHIKNOR W ACFGIJKSY ADKOPQTUY BCFILOQTX

Department of Mathematics University of Leicester Leicester, England LEI 7RH

References [ 11 E.F. Assmus, Jr. and J.H. van Lint, Ovals in projective designs, J. Combin. Theory Ser. A 27 (1979) 307-324. [2] H. Beker and F. Piper, Some designs which admit strong tactical decompositions, J. Combin. Theory Ser. A 22 (15’7) 3842. [3] K . N . Bhattacharya, On a new symmetrical balanced incomplete block design, Bull. Calcutta Math. Scc. 36 (1944) 91-%. [4] N. Biggs, Finite Groups of Automorphisms (Cambridge University Press, London, 1971). [5] P. Dembowski, Finite Geometries (Springer, Berlin, 1%8). [6] W.H. Haemers, Eigenvalue techniques in design and graph theory, Thesis, Techn. Hogeschool Eindhoven, 1979. [7] R. Mathon and A. Rosa, A census of Mendelsohn triple systems of order nine, Ars Combin. 4 (1977) 309-315. [8] E.J. Morgan, Some small quasi-multiple designs, Ars Combin. 3 (1977) 33-23). [9] E.J. Morgan, Arcs in block designs, A n Combin. 4 (1977) 3-16. [ 101 D.P. Shaver, Construction of (0,k, A)-configurations using a non-enumerative search technique, Thesis, Syracuse University, 1973.

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Annals of Discrete Mathematics 15 (1982) 12%134 0North-Holland Publishing Company

ON PAIRWISE BALANCED BLOCK DESIGNS WITH THE SIZES OF BLOCKS AS UNIFORM AS POSSIBLE P. ERDOS and J. LARSON Dedicated

to

Prof. N.S. Mendelsohn on his 65th birrhday

The Let 141= n, C 4, 1 S i C T. is a partially balanced block design, lAll S .. . C 4 1= n"+ O(n"-') for some c >O. authors prove that there is such a design for which 1 If certain plausible assumptions on the difference of consecutive primes are made. then the above inequality can be improved to ]Ail = nI. + O((log np). It is true that there is a design with lAll> nI. - c ? This challenging problem is left open.

Let IS/ = n, Ai C S, 1S i S rn, 2 S IAil < n. Assume that every pair (x, y ) of elements of S is contained in one and only one Ai . A well-known theorem of de Bruijn and Erdos [ 11 states that then rn 3 n where the equality holds if and only if 1A.I = n - 1, (Ail= 2, 1C i < n, or if the Ai are the lines of a finite geometry. Such a geometry can only exist if n = u2+ u + 1, lAil = u + 1. Its existence has been established only if u is a prime or a power of a prime. It is one of the outstanding problems of combinatorial mathematics to prove (or disprove) that such a system can only exist if u = Pa.Here we want t o construct a painvise balanced design which in some sense is as close t o a finite geometry as possible. In fact we prove the following theorem.

Theorem 1. There is an absolute constant c so that for every sufficiently large n there is a painvise balanced design for IS1 = n with the blocks Ai C S satisfying

We will give two proofs for Theorem 1, the first one is constructive and the second one probabilistic which in some sense is more illuminating. But before we prove Theorem 1 we make a few remarks and state some open problems. First of all observe that (1) implies

To show (2), observe that, since every pair of elements of S must be 129

P. Erdos. J.

130

Larson

contained in one and only one Ai, we have

and thus the upper bound of (2) immediately follows from (1). T h e lower bound follows from the theorem of de Bruijn and Erdos. The following problem is interesting but seems difficult: Does there exist a painvise balanced design satisfying

If (3) holds, then as in (2) we would have n 6 m < n + clnlR. At the moment we d o not see how to decide (3), but we will show that if we make certain plausible (but hopeless) assumptions on the difference of consecutive primes, then we obtain the slightly weaker

Constructive proof of Theorem 1. Let pk be the smallest prime for which p i pk + 1 3 n. A well-known theorem of Iwaniec and Heath-Brown [3] states

+

that, for k > k0(&),

Eq. (5) implies that if

is the smallest prime for which p i

+ pk + 1 2 n, then

Let now (Sll= p i + pk + 1 and consider a finite projective Desarguesian plane on S1. Let L , , . . . , Lp:+m+,r lLil = pk + 1 be the lines of S1 and let C b e a conic of our geometry. Let x be a point not on C and L I ,. . . , Lpk+l the lines through x. let further L1,. . . , L(pk-1)/2 be the lines which do not meet C. Put

and by (6), 0 s r < Omit now from s1 the lines L , , . . . , L, and all the r pk + 1 points on it and also omit s points of our conic C (3 conic has pk + 1 points). Thus we are left with a set S of n elements. The lines L , , . . . , L, disappeared, if r < j =s p i + pk + 1, then we now determine how many points we omitted from Lb If

On painvise balanced block designs

131

r < j Spk + 1, i.e., if x E Lj, then we omitted one, two or three points of L,. To see this observe that x has been omitted and if Lj does not meet C we only omitted one of its points. If it meets C, then perhaps one or two more of its points have been omitted. If x Lj (or pk + 2 s j =sp i + pk + I), then we certainly omitted at least r points from L, (since it meets each of the lines Li, I C i S r in one point) and perhaps we omitted one or two of the points L, n C. Let us now denote by what remains from Lj after omitting our points, ( r <j S p i + pk + 1). The sets Al, . . . Ap:+pk+l-r clearly give a pairwise balanced design of the set S, IS1 = n and there are at most six possible values of ] A j ( , namely

This completes the proof of Theorem 1.

Probabilistic proof of Theorem 1. We shall show that if we omit from S1in all possible ways Tn = P i

pk

1-n

elements, we almost surely are left with a set S, which will satisfy (1). We can omit T, elements from S1 in

ways. To complete our proof it will suffice to show that for all but 0((Pi+{+1))

of these omissions, we omitted from each Li, 1 S i S p:

+ pk + 1,

elements. Eq. (7) easily follows by standard methods of elementary theory of probability and we only outline the proof. Put

P.Erdos. 1.Larson

132

Then the number of ways we can omit from p i + pk + 1 elements T,, of them so that there should be at least one line L ; , 1 S i S p i + pk + 1 from which w e omitted more than u or fewer than u elements is clearly less than

A simple computation which we suppress gives that the expression in (8) is

o( ( P i

+f

+

1))

which again completes the proof of Theorem 1. At the moment we do not see how to prove (or disprove) (3). The constructive proof of Theorem 1 gave a pairwise balanced design with only 6 different sizes of the blocks. It would be of some interest to show that 6 can be decreased to 3 and perhaps even to 2. Now we deduce (4) from conjectures on pk+l- pk. The Riemann hypothesis would imply pk+I - pk < P:/2+Eand nearly 100 years ago Piltz conjectured A+!- pk = o(pE). Finally 50 years ago Crarner [2] conjectured that

lim sup(pk,,-pk)/(log k y

=

1.

(9)

Eq. (9) seems to be unattackable by the techniques at QQS disposal. We now deduce (4) from (9). First we prove the following lemma.

Lemma 2. In every finite geometry of p2 + p + 1 points there always is a set of lines L , , . . . L,, r 2 p1/5so that no three of the L, are concurrent and no three of the (;) points Li n L,, 1 == i c j s r are on a line. Proof. The proof of Lemma 2 is simple. Let L I ,. . . , L, be a maximal system of

lines satisfying the conditions of Lemma 2. In other words if Lu is any of the other p2+ p + 1 - r lines of our geometry L, either goes through one of the (;) points Li n L,, 1 s i < J s r or for some k , il, j , , i2, j 2 , 1 S k s r, 1 s i l < j , ZG r, 1 =G i2 < j 2 =z r t h e points L, n Lk, Li,n Lj,, L h f lLiz are on a line. The first condition eliminates at most (ZXp + 1) lines and the second condition

On painvise balanced block designs

133

lines. Thus by our maximality condition we must have

or r > P’’~which proves Lemma 2. Let C be a conic of our geometry. Observe that Lemma 2 remains true if we . . . , L, intersect. The proof of this follows further insist that none of our lines L1, immediately from the fact that there are P’+P+l-(P+l)-(

P 2+ l

)=@

lines not intersecting C. Now we are ready t o deduce (4)from (9). Let as in the proof of Theorem 1p k be the smallest prime for which pi + p k + 1 3 n. Eq. (9) implies that for n > no

Let r be the largest integer for which

and put

and by (10) r S3(log ny. Let now lSll = p i + pk + 1 be a finite geometry and L1,. . . ,Lr+2are r + 2 Iines which satisfy Lemma 2 and do not meet the conic C. Omit the lines Lr,. . . , L,+2and all the points on them and also s points of the conic C. Then we are left with a pairwise balanced design on S, IS1 = n with p:

+ pk - r - 1= n + O(n’’2(log ny)

blocks Ai, 1d i s p i + p k - r - 1. By Lemma 2 a line Lj,j # 1 , 2 , . . . , r + 2 meets Liin at most r + 2 and at least r points, further Lj can meet C in 0, 1 or 2 points. Thus the possible values of ]Ail are

uf:

P. Erdos, 1. Larson

133

which by (10) proves (4). Our method is quite inadequate for the proof of (3) and if (3) is true a new idea will probably be required. The following problem is perhaps of some interest. Consider a finite geometry of n = u 2 + u + 1 points. Let xl,. . . , xk be a maximal set of points no three of which are on a line. In other words the lines joining xi and xi, 1 i < j k contain all the points of our geometry. Determine or estimate the smallest possible value of k. Clearly k > n'I4. Is k = o(nln)possible ? Can the exponent in Lemma 2 be improved '? University of Florida Gainesville Florida, USA

References (11 N.G. de Bruijn and P. Erdos. On a combinatorial problem, Nederl. Akad. Wet. 51 (1948) 1277-1 279.

[2] H. Cramer. On the order of magnitude of the difference between consecutive prime number, Acta Arith. 2 (19.36) 2 . M . [3] D.R. Heath-Brown and H. Iwaniec, On the difference between consecutive primes. Bull. Amer. Math. Soc. (N.S.) 1 (1979) 7S%7600.

Annals of Discrete Mathematics 15 (1982)135-141 @ North-Holland Publishing Company

FINITE REPRESENTATIONS OF TWO-VARIABLE IDENTITIES

OR WHY ARE FINITE FIELDS IMPORTANT IN COMBINATORICS ? Trevor EVANS* Dedicated to N.S. Mendelsohn on his 65th birthday

1. Introduction

It is part of the folklore of algebraic combinatorics that ‘most’ two-variable groupoid identities have non-trivial models in finite fields, the groupoid operation being represented by a linear function ax+ by. The usefulness of this result stems from the fact that two-variable identities (in one or more binary operations) are frequently a natural tool for the algebraic description of combinatorial structures. For example, the variety of quasigroups (the algebraic analogue of latin squares) may be defined by three binary operations and four two-variable identities. Many particular varieties of quasigroups which occur in combinatorial contexts are either defined by further two-variable identities or sometimes may be characterized as a variety of groupoids (one binary operation) satisfying some two-variable identities. The variety of Steiner quasigroups (corresponding to Steiner Triple Systems) is an example of this. Cyclic decompositions of graphs also give rise to varieties of groupoids defined by twovariable identities and k-sets of m.o.1.s. may be regarded as algebras in a variety defined by (a large number of) binary operations and two-variable identities. As far as I know, the first (partial) answer to the question whether any two-variable groupoid identity (with the obvious exception of w ( x ) = y) has a non-trivial finite model was given by Saade [8].Shortly after this Austin [ 11 published a complete (affirmative) answer to the question showing that any non-trivial groupoid identity in two variables has a model of the form x * y = ax + b y on Z k, the integers (mod k), for some k > 1. In a survey article [7] N.S. Mendelsohn outlined a proof showing that one can obtain for quasigroup This research was supported in part by NSF Grant MCS-7903693. 135

136

T. Evans

identities in two variables a model in a finite field, again with the quasigroup multiplication given by a linear expression i n ,the field. He also mentions that one can obtain a similar result for varieties of'gempotent groupoids. The original purpose of this note was two-fold. In the first part, which dates back more or less to the time of [8], we give a simple proof that any non-trivial groupoid identity which does not have as models the uninteresting cases of constant or left- or right-zero semigroups necessarily has infinitely many idempotent quasigroup models on finite fields of the form x * y = (1 - a)x + ay. It follows that any groupoid identity in two variables with non-trivial models has models on finite fields. It is easy to extend these results t o a two-variable identity involving any finite number of finitary (non-nullary) operations, as Austin has observed [ 11. The second part of the note was to be an interpretation of these results in terms of clones and is actually the reason for the author's renewed interest in the problem. Clones given by generators and relations correspond t o varieties and it is natural to look at the simplest case first, namely 2-clones on one generator and defined by one relation. But these are nothing more than groupoid varieties defined by one identity in two variables. Thus, the results of this note may be interpreted in terms of representations of clones on finite rings. However, the ramifications of these ideas become so technical and extensive that it seems better to leave them for later publication. Some indication of these ideas is given in the survey paper [4].

2. The triviality problem for two-variable identities Consider a variety Sr defined by a two-variable identity in one binary operation

When is 7f non-trivial, i.e., contains non-trivial algebras ? Clearly if both u and 11 are words of length greater than one, then any groupoid with a constant multiplication lies in Sr. To exclude such models, let us restrict our attention t o varieties defined by identities u = u where u is of length one, i.e., consists of a single variable y. i f u(x, y) begins with y , then any left-zero semigroup (satisfying xy = x) lies in Y' and similarly if u(x, y ) ends in y, any right-zero semigroup (satisfying x y = y ) lies in Sr:, Hence, in deciding whether u(x, y ) = y has non-trivial models, we need only consider the case where u(x, y) begins and ends with x. If u(x, y) contains n o occurrence of y. then clearly u(x, y) = y implies x = y and

137

Finite representations of two-variable identities

'=V is trivial. In fact, as we shall see, w ( x ) = y is the only type of two-variable identity which does not have non-trivial models. We have reduced the problem now to considering identities of the form w(x7

Y)=Y

where w ( x , y) begins and ends with x and contains at least one occurrence of y. The following sequence of lemmas enables us to show that the identity actually has idernpotent quasigroup models on infinitely many finite fields. Each proof is either trivial or a simple induction on the complexity of the groupoid word (i.e., the number of occurrences of the groupoid operation in. the word). Let (x, y) be a groupoid word in the binary operation ( and variables x, y, and let z[r]be the ring of polynomials in r with integer coefficients. If we represent the groupoid operation x * y as a linear function a )

(1 - r)x

+ ry

on H I T ] , then the derived operation in x and y given by w(x, y) corresponds to the linear expression G ( x , y) on Z [ r ] constructed recursively by the rule (i) f = x, = y, (ii) if w ( x , y ) = u ( x , y ) v(x, y), then G(x, y) = (1 - r ) x w ,Y)+ m x , Y). We may also describe G(x, y) directly, as a linear expression in x and y in terms of a polynomial w ( r ) in Z [ r ] defined by: (i) if w ( x , y) = x, then w ( r ) = 0 ; (ii) if w ( x , y) = y, then $ ( I ) = 1 ; (iii) if w ( x , y) = u ( x , y ) * u(x, y), then G ( r ) = (1 - r ) d ( r )+ 6 ( r ) .

-

Lemma 1. @(x, y ) = (1 - C ( r ) ) x

+ G(r)y.

Proof. By induction on the complexity of w(x, y ) . If w(x, y) = u(x, y ) * v(x, y), then by the definition of G(x, y) and the inductive assumption

x , Y ) + ~ ( x Y, ) (1 - rX(1- ii(r))x + ii(r)y} + r((1- f ( r ) ) x + f ( r ) y } = (1 - ((1 - r)C(r)+ 6 ( r ) } ) x + ((1- r)C(r) + 6 ( r ) ) y = (1 - $(r))x + $(r)y.

w ( x , y ) = (1 - M =

Lemma 2. If w(x, y ) is a groupoid word beginning with x, then G(r) has r as Q factor.

Proof. If w ( x , y ) is x, then a(r)= 0. If w(x, y) = u(x, y ) v(x, y ) , then G(r) = (I - r)ii(r)+ G(T).By the induction hypothesis d ( r ) has r as a factor. Hence, G ( r ) has r as a factor.

7.Ewns

138

Lemma 3. If w(x. y ) ends in x, then w ( r ) has 1 - r as a factor. Lemma 4. If w(x, y ) contains no occurrence of y, then G ( r )= 0. Lemma 5. If w(x, y ) contains at least one occurrence of y , then G ( r ) # 0. Prod. We prove by induction that the coefficient of the term of lowest degree in G ( r ) is positive. If w(x. y ) is y, then w ( r ) is 1. If w(x, y ) = u(x, y ) . v(x, y ) , then G ( r ) = (1 - r ) i i ( r )+ $ ( r ) . At least one of u(x, y ) , v(x, y ) contains an Occurrence of y and so by induction each of ii(r), B(r) is either 0 or has its term of lowest degree positive. Hence, w ( r ) has this property also.

Lemma 6. If w(x. y ) is a groupoid expression beginning and ending with x and containing at least one occurrence of y, then G ( r ) = r(1- r ) q ( r ) where q ( r ) is a non -zero polynomial over the integers. We can now state the main theorem. Theorem 7. Every groupoid idenriry

w(x. y ) = ?-' where w(x, y ) begins and ends in x and has at least one occurrence of y in it, has a non-trivial idempotent quasigroup model on some finite field F. The groupoid multiplication x . v in the model on F is given by x .y

= ( I - a)x + a y

where a is an element of F ( # 0, 1) which satisfies the equation G ( r ) = 1.

Proof. Let GF(p) be any finite prime field in which G ( r ) = r(1 - r)q(r) is not identically zero, i.e., not all coefficients of q ( r ) divisible by p . Then either there is an element a # 0. 1 in GF(p) such that a(1- a ) q ( a )= 1 or there is such an element in a finite extension of GF(p). By Lemma 1, if we take the groupoid operation x . y as (1 - a ) x + ay, the identity w(x, y ) = y will be satisfied by this operation. Example. Consider the identity (XY

.Y)X

=

Y.

Here, w(x. y ) = (xy . y)x, G ( r ) = ?- 3r2 + 2r. The equation C ( r ) = 1 has a solution r = 3 in GF(5) and so we may obtain a model by taking the groupoid operation to be 3 x + 3 y in GF(5). We can also extend GF(2) by a root of r3 + r + 1 and obtain a model on GF(23).

Finife representations of two-variable identities

139

Remarks. (1) It follows from the above theorem that every groupoid identity in two variables, with the exception of those of type w ( x ) = y, has a model, with the groupoid operation represented as a x + by, on infinitely many finite fields. The constant models are given by a = b = 0 and the left- and right-zero models by a = 1, b = 0 and a = 0, b = 1. (2) The same techniques used to prove the theorem may be applied to the groupoid identities which we disregarded because they had constant models or left- or right-zero semigroup models. Of course, an idempotent quasigroup model on some finite field of such an identity may not be found this way because it may not exist. Here are some examples of the different situations which can occur. (i) The identity x y * y = yx . xy, which we know has a right-zero model of every order, yields the equation (1- r ) r + r . 1 = (I - rX(1- r ) 1 + r .O}+ r((1- r ) O + r * 1)

i.e., 3r2 - 4r + 1 = 0 or ( r - 1)(3r - 1) = 0. The solution r = 1 corresponds to the right-zero semigroup models. For another solution, over any GF(p), p f 3, we can solve 3r - 1 = 0 and obtain a model. GF(7) is a model, with x . y defined as 3x + 5 y . (ii) The identity x . yx = x y . x, which has constant and left- and right-zero semigroup models of every order, yields the equation r(1- r ) = (1 - r)r or 0 = 0. This reflects the fact that, in any finite field, and for any element a in the field, the operation x * y = (1 - a)x + a y satisfies this identity. (iii) The identity x y . x = y x xy does not have any left- or right-zero semigroup models. We obtain from it the equation 3r2- 3r + 1 = 0. Thus, we do have, in this case, infinitely many idempotent quasigroup models. GF(7) is a model with the operation defined as 6x + 2y. (iv) The identity x y yx = x has n o constant multiplication model but does have left- and right-zero semigroup models of every order. The equation in r we obtain is 241 - r ) = 0 showing r = 0 and r = 1 left- and right-zero semigroup models. We might suspect from this equation that there is also a model in a field of characteristic two. This is indeed the case, with (1 - r ) x + ry as an idempotent quasigroup model in GF(22), where r is a root of 9+ r + 1 in the extension field of GF(2). (3) It is easy to deduce, from the groupoid result, that any variety defined by a two-variable identity and finitely many n-ary operations, n 2 1, has nontrivial finite models (again, with the obvious exceptions, w ( x ) = y ) . Rather than prove this, we give an example. The constant operation models and the analogues of the left- and right-zero semigroup models are treated as before. Consider the identity

-

-

f(&,

y , Y ) , W ) ,V Y , x),

QX,

Y, Y, X I ) = Y

T. Evans

140

where h is unary, k binary, g ternary, f and 1 quaternary operations. In this equation replace every unary operation by the identity operation and every n-ary operation, for n > 1, by some composition of a binary operation x y . For example. replacing f ( x , y , z, w ) by ( x y . z ) w , g(x, y, z) by x y . t, k ( x , y ) by x . y. l(x. y , z , w ) by x y t w , we obtain

We obtain a model of the form (1 - a)x + a y for this multiplication on some finite field and define t h e representations off, g, k, 1 in terms of this, to obtain a model for t h e original identity in f , g, k, 1. (4) Since two-variable identities prove to be so tractable, it is natural to look next at three-variable groupoid identities. Here, the situation changes drastically. For example, none of the identities x ( y * y z ) = y , x ( y y * z ) = y , (x . y y ) z = y , ( x v * y ) z = y have non-trivial models but x y . y z = y does, although they are not so easy to find [3]. There are fairly simple three-variable identities which have n o non-trivial finite models but which do have infinite models, e.g. [2]. Probably the simplest interesting three-variable groupoid identity is xy . y z = y mentioned above. The author first looked at it for purely algebraic reasons, Knuth [5] showed that its study is equivalent to that of directed graphs having a unique path of length two between any two vertices and also to solutions of the 0, 1-matrix equation X 2= J. and N.S. Mendelsohn [6] generalized this, describing a variety of groupoids equivalent to graphs having unique paths of length n between any two vertices and also to solutions of the equation X" = J. Unfortunately, it is not possible to represent groupoid identities in three variables using linear polynomials over finite fields-the identity x y . y z = y is an example of this. Of course, this identity may be modelled in some finite fields (those of square order) using a non-linear polynomial in x and y for the groupoid operation. Department of Mathematics and Computer Science Emory University Atlanta, U S A .

References [ I ] A.K. Austin. Finite models for laws in two variables, Proc. Amer. Math. Soc. 17 (1%) 14 10- 14 12.

12) A.K. Austin, A note on models of identities, Proc. h e r . Math. Soc. 16 (1965) 522-523. [3] T. Evans, Products of points--some simple algebras and their identities, Amer. Math. Monthly 74(4) (1967) 362-372.

Finite representations of two-variable identities

141

[4] T. Evans, Some remarks on the general theory of clones, Roc. Conf. Finite Algebra and

Multiple-valued Logic (North-Holland, Amsterdam, 1982).

[5] D.E. Knuth, Notes on central groupoids, J. Combin. Theory 8 (1970) 376-390.

[6] N.S. Mendelsohn, Directed graphs with the unique path property, Combinatorial Theory and its Applications I1 (North-Holland, Amsterdam, 1970) pp. 783-799. [7] N.S. Mendelsohn, Algebraic construction of combinatorial designs, Roc. Conf. on Algebraic Aspects of Combinatorics, Toronto, 1975, 157-168. [8] M. Saade, Some problems concerning identities in algebras, Ph.D. Thesis, Emory University, 1966, chapter 4.

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Annals of Discrete Mathematics 15 (1982) 143-159 @ North-Holland Publishing Company

SOME CONNECTIONS BETWEEN STEINER SYSTEMS AND SELF-CONJUGATE SETS OF M.O.L.S.

T. EVANS* and M. FRANCEL To Nathan Mendelsohn on the occasion of his 65th birthday

0. Introduction

In [9] Ganter and Werner described a generalization of the well-known representation of a Steiner triple system as an idempotent totally symmetric quasigroup. They constructed a variety of algebras in which each algebra is a Steiner (t, k)-system, the blocks (of size k) being the t-generator subalgebras. However, this algebraic version of a Steiner system can only be obtained (excluding trivial cases) if t = 2 and k is a prime power or if t = 3, k = 4. In [ll] Lindner and Mendelsohn extended the notion of a conjugate of a quasigroup to that of a conjugate of an n2X k orthogonal array, obtained by permuting the columns. The conjugate invariant subgroup of an array is the group of all permutations which yield conjugates equal to the original array. In [lo, 11, 121 the cases k = 3, 4 are covered. They characterize the groups which can be conjugate invariant subgroups for n2 X 3 and n 2X 4 orthogonal arrays and describe in each case the identities satisfied by the quasigroups determined by the arrays. The main purpose of this paper is to describe a rather unexpected connection between the ideas described in the preceding paragraphs and illustrate the use of some algebraic tools developed by one of the authors to study sets of m.o.1.s. and two-variable identities. Certain groups are associated with sets of m.o.1.s. and these groups turn out t o be closely connected with the Lindner-Mendelsohn conjugate invariant subgroups. In one of the most interesting cases, where the group is ‘aslarge as possible’, the corresponding set of m.o.1.s. determines a Steiner (2, k)-system and the Ganter-Werner algebra associated with it. Although the algebraic tools are described elsewhere ([3, 51 and forthcoming papers), enough of the general theory is given t o make this paper selfcontained. In another paper by one of the authors ([7]; see also [S]) the The work of this author was supported in part by NSF Grant MCS-7903693. 143

T. Evans, M.Francel

144

approach described here is used t o study results corresponding t o the n 2 x 3, n2 x 4 cases discussed by Lindner et al. mentioned above and to extend these results to other values of k.

1. Algebras of tables

Let N = { 1 , 2 , . . . , n } and let 58 denote the set of all binary operations a :(x, y)+ a(x, y) on N. We will often identify a with its table, i.e., the n x n array having a(x. y) in its x-row, y-column. We define on 9 a ternary operation of composition [a, b, c], where, for any a, b, c E 58, [ a , b , c I : ( x , ~ ) ~ a ( b ( x , y ) , c ( x , yX) ,)Y,E N .

(1.1)

Under this operation 9 is a clone of functions satisfying, for all a, b, c, d, e in 3,

[a,pi, pz] = a

I

[PI,a, b] = a

b2,

a,b ] = b y

(14

where pI, p? are the projection operations given by p,(x, y) = x, pr(x, y) = y, This is called the generalized associative law. On 9X 8,we define a monoid M by (a, b ) . (c, d ) = ([a. C, 4,[b, C,

4 ).

(1.4)

The neutral element of “44 is (Pl,p2).The subgroup of all invertible elements in A will be denoted by 9. Note that A! is the monoid of all mappings N 2 + N 2 and c$ is the group of all permutations on N2. Identities in two variables satisfied by an operation a in 9 correspond to equations involving a in the clone and the monoid A. For example, the commutative law a(x, y) = a(y, x) corresponds to a = [a, p2,p,] in 9 and the law a(x, a(x. y)) = y (in multiplicative notation x * xy = y) corresponds to the clone equation [a,pl, a] = pr or, more elegantly, t o the monoid equation (PI, a)’ = (PI.p2) in A. For more examples of this and information about clones, we refer to [5, 61.

2. Orthogonal tables

Two n x n tables al, a2 in 58 are orthogonal, written a1 IU Z , if the mapping (x, y ) 4 (al(x,y), a2(x,y)), x, y E N, is a bijection, i.e., superimposing one table o n the other results in an n x n array of ordered pairs of elements of N in which each ordered pair in N 2 occurs exactly once. We write I { a l , a?,. . . , ak}

Some connections between Steiner systems and sets of m.o.1.s.

145

if a i l aj for all i f j , and call { a l ,a2,. . . , ak} a set of mutually orthogonal tables (or groupoids). Lemma 2.1. al Ia2 if and only if (al, a2) E 3, i.e., ( a l , a2) is invertible in A. Lemma 2.2. a is a latin square if and only if

(PI,

a ) and (a,p2) are in $7.

Proof. a is a latin square if and only if it is orthogonal to both p 1 and p2. Lemma 2.3. If {al, a2,. . . , ak} is a set of mutually orthogonal tubles and al = pl, a2= p2, then {a3,a4,. . . , ak} is a set of m.o.1.s. A set of mutually orthogonal tables { a l , a 2 , .. . ,ak} in 8 such that two of them are the projections pl, pz (and hence the others are latin squares) will be said to be normalized. In a useful alternative terminology we will refer to the result of adding the projections P I , p2 to a set of m.o.1.s. as an augmented set of m.o.1.s. When we refer to a normalized set of orthogonal tables { a l ,a 2 , .. . , at} or to an augmented set of rn.o.1.s. {al,a2, . . . , uk} without further details, it is to be always assumed that al is p 1 and a2 is pz. More details on this algebraic view of orthogonality are given in [3, 51.

3. Conjugates of sets of miitually orthogonal tables

Let d = {a,,a2,. . . , ak} be a k-set of mutually orthogonal tables on N = {1,2,. . . , n} and let (u, v ) E %. Then the set d'={ui,a;, . . . , a 2 where a : = [ai,u, u ] , i = 1,2,. . . , k, is also a k-set of mutually orthogonal tables on N since ([a;,y v ] , [a,, u, u ] ) = (ai, aj)(u, u ) in 9. We call d' a conjugate of d and write d'= d ( u , u). Note that (u, u ) acts on d by rearranging the entries of each table a; in d by the permutation (u, u): N 2 + N 2 given by (x, Y ) - + ( 4 x 7 Y ) , u(x9 Y ) ) . Lemma 3.1. Conjugacy is an equivalence relation on the set of all k-sets of mutually orthogonal tables in 9.

Proof. This follows from the properties of the clone operation on 9.Since d(pl, p2) = d,conjugacy is reflexive. Now let d'= { a ; ,a;, . . . , a i } be conjugate to d,i.e., for some (u, u ) E 92, a : = [a;,u, u ] , i = 1,2,. . . , k. If (u, v ) - ' = (s, t), then [a:,s, t] = [[ai,u, 01, s, t ] = [ai, [Y s, tl, [u, s, t ] ] = [ai,pi, p2] = a;. Hence, d = d ' ( s , t ) and conjugacy is symmetric. Similarly, if d'= d ( u l ,ul) and d"= d'(u2, u2), then in d',a: = [al, ul,v,], in d",a9 = [ a ; ,u2, v2]. Hence, a7 = [[a;,ul,ul], u2, v2]= [a;,[ul, u2, v2],[ V I , u2, u2]]. That is, gll=

T. Evans, M.Francel

146 ~ ( [ U I ,uz, UZ].[UI,u2, U Z ] ) =

s8((ul, uI)(u2, u2)) and so conjugacy is transitive.

It follows immediately from the definition that any two pairs of mutually orthogonal tables are conjugate and in particular that any pair of orthogonal tables is conjugate to {pl,h}.A generalization of this gives an extremely useful result.

Lemma 3.2. Any k-set of mutually orthogonal tables is conjugate to a normalized set. Proof. If d = {al,u 2 , .. . , a r } is a set of mutually orthogonal tables, then .$(a,, uj)-' contains pl, p2 since if (a;,aj)-l = (u, u), then [a,, y v ] = p l , [a,, u. U J = p2. A sort of converse of this provides our main tool for studying subgroups of 9 which leave invariant given sets of mutually orthogonal tables.

Lemma 3.3. If d = {al, u2,. . . , ak}, and its conjugate d ( u , u ) is a normalized set of mutually orthogonal tables, then (u, u ) = (a;,ui)-'for some a;, ai in d. proaf. pI, p2

belong to d ( u , u ) and so (a,, aj)(u, u ) = (PI,p2)for some a;, aj in d.

Remark. The word conjugate was first used by Stein [14]. If a quasigroup multiplication x . y = z on a set 0 is regarded as a ternary relation {(x, y, z): x . y = z in then applying a permutation to the elements in each triple in the relation gives andther ternary relation on Q which determines a quasigroup operation. These are the conjugates of the original multiplication. The idea goes back to Etherington [l] and Sade [13]. In the notation of Evans [2], the six quasigroup operations we obtain are determined by the original multiplication x * y as xy, x \ y , x / y , yx, y\x, y / x . It seems reasonable to extend the original use of 'conjugate' to our sense since, if a, b are quasigroup operations, then they are conjugate in the sense of Stein if and only if the augmented sets (PI,p2, a}, (PI,p2, b} are conjugate in our sense. Also, if {al. u2,u3} is a set of m.0. groupoids, then the conjugate sets {al, u2,a3}(ai,aj)-', isC j , are of the form {pl,p2, b}, where b is one of six conjugate quasigroup operations.

a},

4. The stabilizer of a set of m.0. tables

Let d = { a l , a2,. . . , ak} be a set of m.0. n x n tables (or groupoids) on { 1 , 2 , .. . , n}. The group $? acts on the set of all conjugates of s8 by

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(u, v ) : d +d ( u , u), where (u, v ) E %. In particular, some elements of % map d onto itself under this action. We define the stabilizer of d,written stab d,to be the subgroup of % consisting of all (u, v ) in % such that d ( u , v ) = d.We collect

together in this section some useful properties of the stabilizer.

Lemma 4.1. Every element of stab d is of the form ( a l ,az)-'(4, aj), i # j . Proof. If (u, u ) E stab d,then d ( u , v ) = d and so [al, u, v ] = ai, [az,u, u ] = aj for some if j . That is, (al, az)(u,u ) = (ai, aj) and so (u, u ) = (al, a2)-'(ai,aj).

Lemma 4.2. If d is a normalized set of m.0. fables, then every element of stab d is of the form (ai,aj),i # j . Proof. In this case, (al, a 2 ) =(pl,pz).Thus, for normalized sets of m.0. tables d,we have a very simple description of the elements in stab d.

Theorem 4.3. If d and 9 = d ( s , t ) are conjugate sets of m.0. tables, then their stabilizers are conjugate subgroups of 9, in fact, stab $53 = (s, t)-'(stab d ) ( s , t) . Proof. Since 58 = d ( s , t ) , if (u, v ) E stab d,then 9 ( s , t)-'(u, u)(s, t ) = d ( u , u)(s, t ) = d(s,t ) = 58.

(s, t)-'(u, u)(s, f ) E stab 9. Conversely, if (u', v ' ) E stab 58, then d ( s , t)(u', v ' ) = d ( s , f ) , i.e., (s, t)(u', u')(s, t)-' E stab d. Hence, (u', v ' ) = (s, t)-'(u, u)(s, I ) , for some (u, v ) in stab d.

Hence,

Example. We may illustrate the ideas so far and, in particular, those of this section, by describing the possible stabilizers for d = {al,az,a3},a set of three m.0. groupoids. First, we note that the conjugate 3 = d(a1, a*)-' has the form

where a is a quasigroup operation, which, if we write it as x - y , may be described either by

where (b, c ) = (al, a&', or by

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Since stab D = (al,a2)(stab&)(al,4’by Theorem 4.3, we may restrict ourselves to considering stab 9. This is made easy by Lemma 4.2. The only elements of 9 which are candidates for mapping 9 = {pl.pr, a} onto itself are ( P I , P2). ( P 2 , PI), ( P I . a ) , (a,PI), 0-727 a ) , ( a ,P2). (i) Stab 9 contains (P2,pI) if and only if {PI,pz, a} = {PI,p2. aI(p2, PI)= (Pz, PI,[a, pz, PI]}. that is, a = [a, p2, p l ] ; a satisfies xy = yx, (ii) Stab 3 contains (pl, a ) if and only if {PI, PZ,a} = {PI, p2, a}(pl, a ) = {pl,a, [a, pI,a]}, that is, p2 = [a, pl, a]; a satisfies y = x . xy. (iii) Stab 98 contains (a, p l ) if and only if {PI,p2, a} = (PI,p2, a}(a,PI)= {a,p l , [a, a , p l ] } ,that is, pz = [a, a, p l ] ; a satisfies y = xy * x. (iv) Stab 3 contains (p2,a) if and only if (PI,p ~a} , = {PI,p2, a } h , a ) = {h.u, [a, pz, a]}, that is, p1 = [a, pz, a]; a satisfies x = y * xy. (v) Stab 9 contains (a, p2) if and only if (PI,p2, a} = {PI,pz, a}(a,pz) = {a.p ~[a, , a, p 2 ] } ,that is, pl = [a, a, p2];a satisfies x = x y . y. Each candidate for membership in the stabilizer of 3 (other than (p1,p2)) corresponds to an identity satisfied by the quasigroup operation a ( x , y). Now we consider the possibilities for the actual group, stab 9. We use the conditions listed in (i) through (v) above. (1) If stab 9 contains (p2,p,), then the group {(PI,p2),(P2,pl)} is included in stab 3. (2) If stab B contains (PI,a ) , then p 2 = [ a , p l ,a] by (ii) and (PI, a)2= (pl, [a,PI, a ] )= (PI,p2). Hence, the group {(PI,p2),(PI,a)} is included in stab 9. (3) Similarly, if stab 9 contains (ah),then (a, p r y = (pl,p2) by (v) and {(pl,p2),(a, pZ)} is a group included in stab 9. (4) If stab 9 contains (a. p l ) or (P2, a), then by (iii), (iv) it contains both, since (a,PI)’ = @2, a ) , @2. a y = (a,PI). Hence, {(PI,PZ),(a,PI), (p2, a)} is a group included in stab 9. (5) If stab B contains any two of (pz, pl), (PI, a ) , (a, p2) or any one of these and one of (a, pl), (p2,a), then it contains all five and the group {(PI.pz), (P2, pl), (PI, a ) , (pz, a),(a, PI), (a,pz)} is included in stab 3, in fact, is stab 8. Hence, the possible non-trivial stabilizers for 9, are (a) {@I, pz), (Pr, p)}: in this case a satisfies x y = yx but does not satisfy s . xy = y ; (b) {(PI,p?),(PI, a)}: in this case a satisfies x xy = y but does not satisfy xy = y x :

-

(c) {(PI.pz), (a, p2)}: in this case a satisfies xy . y = x but does not satisfy

xy = y x :

-

(d) {(pl, p2), (a,PI),(p2,a)}: in this case a satisfies xy . x = y or x yx = y (these are equivalent) but does not satisfy x y = y x ;

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(e) {(PI, pz), (pz, P I ) , @I, a ) , (a,P I ) , (a,P A(pz, a)}:in this case a satisfies all of the identities x y = yx, xy y = x, x * x y = y , x y y, x * yx = y, that is, a is a totally symmetric quasigroup. If 93 has only a trivial stabilizer, then the operation Q ( X , y ) does not satisfy any of the above identities, and conversely. It will be clear after we have introduced the notion of an array stabilizer, that the above results are equivalent to those obtained by Lindner and Steedley [lo].

-

-

5. The stabilizer as a permutation group Since each (u, u ) may be regarded as a permutation

on N x N, the stabilizer of a set d = (al,az, . . . , at} of m.0. tables may be thought of as a permutation group on N x N. However, another representation, as a permutation group on {1,2, . . . , k} turns out to be much more useful. We define, for each (u, u)E stab d, a permutation 0,” on {1,2, . . . , k} by

Theorem 5.1. The set of all such 0,, is a permutation group isomorphic to stab d. Proof. The mapping (u, u)+ 0,” is one-one since if 0,” = flu.,,,,then [ui, u, u ] = [ai,u’. 0’1 for all i. Hence, for i# j , (ai,aj)(u, u ) = (ai,a,)(u’, u‘) in 97, i.e., (u, u ) = (u’, u’). The mapping is obviously onto. For the homomorphism (u’,u’)+ 0,~,~. Then (y u ) (u’, u‘) = property, consider (u, u)+ ([y u‘, u’], [u, u’, u‘]) and this maps onto cp say, where uirp= [ai, [u,u’,0’1, [u,u’,u’]]= [[ai, U, u ] , u’,u‘] = a i w .

-

We will call this permutation representation of stab d, the array stabilizer of d ; the reason for this terminology will become apparent in the following

section. Clearly, if Sa, 93 are conjugate sets of m.0. tables, then their array stabilizers are isomorphic. However, we can say much more than this.

Theorem 5.2. If d = {al, a,, . . . , ak}, 93 array stab d = array stab 93.

= { b, ,b3,.

. . , bk} are conjugate, then

T.Evans. M. Francel

I50

Proof. Let 93 = d ( c , d) and let 8 E array stab d.Then, for i = 1,2, . . . , k,

which is, by Theorem 4.3, an element in the stabilizer of 58 = {b l,b 2 , .. . , bk}. Hence, 8 belongs to the array stabilizer of 9. Since conjugacy is symmetric, it follows similarly that a permutation is in array stab 9 if and only if it is in array stab d.

Remark. In terms of the array stabilizer, the example at the end of Section 4 may be restated as follows. If d = { a , ,a2,a3}is a set of m.0. tables conjugate to (PI,p2,a) then the possibilities for the array stabilizer of .& are the subgroups of the symmetric group on {1,2,3}. Each subgroup corresponds to a certain set of identities satisfied by the quasigroup operation a(x, y ) . For example, array stab s8 is the symmetric group on { 1,2,3} if and only if a(x, y) (written as x . y ) satisfies x y = yx and xy * y = x. 6. The Lindner-Mendelsohn stabilizer A n n2 x k orthogonal array on N = {1,2,. . . , n } is usually defined to be a

rectangular array of n2 rows and k columns where, for any two distinct columns, the set of ordered pairs occurring in these two columns and the n2 rows is precisely the set of all distinct n2 ordered pairs from N. A slightly different definition used by Lindner and Mendelsohn [ll] and which we will also,-use here, essentially identifies any two orthogonal arrays in the above sense if they differ only by a permutation of the rows. Thus, an n Z x k orthogonal array on N is a set of n2 ordered k-tuples of elements of N, {(zy, z:,

. . . , 2:): (i, j ) E N

X

N},

(6.1)

such that for any pair s f t in {I, 2, . . . , k}, the set of all ordered pairs 2;): ( i , j ) N ~ x N} contains all n 2 ordered pairs of elements of N. In this second definition an orthogonal array is simply the set of all rows of an orthogonal array as first defined. {(z;,

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of m.o.1.s.

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If d = {al, a2,. . . , ak}is a set of mutually orthogonal n x n tables on N, then we define the n2 x k orthogonal array O A ( d ) ,associated with d, to be the set of k-tuples

For each (i, j ) in N x N, the corresponding k-tuple is the sequence of elements occupying the (i, j)-celi in the tables ul, a2,. . . , ak. Lemma 6.1. If d,3 are conjugate sets of m.0. tables, then O A ( d )= OA(93).

Proof. Let 9 = d ( u , u), where (y u)E 22. The action of (4 u ) on any ui in st = {al, u2,. . . , ak} consists essentially of permuting the cells and thus corresponds t o a permutation of the elements of the set OA(d). Formally, we note that as,:(i, j ) + (u(i,j), u(i, j ) ) is a permutation on N x N and so the two sets

are equal. (we have written a for as, in OA(93).) Lindner and Mendelsohn define the conjugate inuariant subgroup for an n2 x k orthogonal array W,to be the group of all permutations 8 on 1,2, . . . , k such that 3 0 = W,where W 8 is the orthogonal array obtained from W by permuting the columns of W by 0, i.e., the tth column of 9 becomes the (t8)th column of 9 8 . From Lemma 6.1 it is immediate that if d,98 are conjugate sets of m.0. tables then the conjugate invariant subgroups corresponding t o d and 93 are equal. The main purpose of this section is to show that for any set of m.0. tables d = {al,a2,. . . ,ak}, then the array stabilizer of d and the conjugate invariant subgroup of the orthogonal array O A ( d )are the same.

.

Theorem 6.2. Let d b 2 g s$ of m.0. tables. Then array stab d = the conjugate inuariant subgroup for O A ( d ) . Proof. If d = {al, a2,. . . , ak}, then both groups are subgroups of Y;, the symmetric group on {1,2, . . . , k). Let 8 E array s t a b d . Then, for i = 1 , 2 , . . . , k, aie = [ai, u, u ] for some (4 u ) E 46.

(6-3)

T. Evans. M. Francel

I52

Since O A ( d )= {(a,(i,j ) , a2(i,j ) , . . . , ak(i, j ) ) : (i, j ) E N x N } , [ O N 4 1€9 = {(a,@-l(i, j ) , . . . , ake-I(i,j ) ) : ( i , j ) E N X N } = {(U,e-l(u(i’, j ‘ ) , u(i‘, j ‘ ) ) , . . . , &@-I(u(i‘, j ’ ) , u(i‘,j ’ ) ) ) : (i’, j ‘ ) E N x N }

where (u, u ) : (1‘. j’)-

(i, j ) (we recall that (u, u ) is a permutation of N

X

N)

{([alO-i, u, u](i’, j ’ ) , . . . , [ake-l, u, v](i’,j‘)): (i’, j ’ ) E N x N } = {(a,(i’,j ‘ ) , . . . , ak(i’, j‘)): (i’, j ’ ) E N X N} by (6.3) = O A ( d ). =

Hence, 0 belongs to the conjugate invariant subgroup for O A ( d ) . Conversely, let 8 be a permutation in this subgroup. Then O A ( d )= {(a,((j ) , . . . , ak(i, j ) ) : (i, j ) E N X N}, [OA(d)]O= {(ale-l(i, j ) , . . . , ake-l(i, j ) ) : (i, j ) E N X N ) . Since O A ( d )= [OA(d)]O,there is some permutation a on N may write as

X

N, which we

where u, u are orthogonal binary operations on N, such that for all i, j in N Hence,

u,(i,j ) = a@-ia(i,j ) ,

t = 1,2, . . . , k .

u,(k j ) = uM-1(4i, j ) , u(i, j ) ) , a,(i,j ) = [ure-l,u, u](i,j ) for all i, j in N, t = 1 , 2 , .. . , k . a, = [a,e-l,u, u ] , ure= [a, U, v ] ,

But this last equation states that 8 E array stab d. Hence, the array stabilizer of d is identical with the conjugate invariant subgroup for the array O A ( d ) and from now on we can dispense with this latter term. In a sense, we can also dispense with the term army stabilizer since the elements of stab d can be regarded as permutations acting on {al, az,. . . , ak}, i.e., for (u, u ) E stab d.we define a(&”) by = [ai, u, u ] and this permutation group is clearly isomorphic t o the array stabilizer of d.

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7. Steiner systems

In this section we will consider an augmented set of m.o.1.s. d = { a l ,a2,. . . , ak}whose stabilizer is ‘as large as possible’. Since each element of stab d is of the form ( a , aj), i # j , this happens if stab d consists of all such ordered pairs, hence is of order k(k - 1). Equivalently, since any element of the stabilizer is completely determined by its effect on p1 and p2, stab d is ‘as large as possible’ if, for any a,# a, in d,there is an element in stab d mapping pl onto a,, p2 onto a,. But this is merely saying that the array stabilizer of d is a sharply doubly transitive group on {1,2,. . . , k} (or stab d is a sharply doubly transitive group on { a l ,a2,.. . , ak}).We sum up these remarks in the form of a lemma. Lemma 7.1. The following properties of an augmented set of m.o.1.s. d = { a l ,a2,.. . , ak} are equivalent: (i) Stab a2 is ‘aslarge as possible’. (ii) Stab a2 has order k(k - 1). (iii) Stab SP contains all (ai,aj), i f j . (iv) Array stab 94 is a sharply doubly transitive group on {1,2, . . . , k}. (v) Stab a2 is a sharply doubly transitive group on {al,a2,. . . , ak}.

From now on, even if it is not mentioned explicitly, we will assume that

d = {al,a2,.. . , a k } is an augmented set of m.o.1.s. satisfying the properties described in Lemma 7.1 and that each ai is a table on N = {1,2, . . . , n}. We

need some preliminary results before we can prove one of our main resultsthat d determines a Steiner (2, k)-system of order n.

lheorem 7.2. I f { a l ,a2,. . . , at} is an augmented set of m.o.1.s. which has a sharply doubly transitive stabilizer and k 2 4, then each ai is idempotent.

Proof. Consider the mapping a :{3,4, . . . , k}+{l, 2,3, . . . , k} given by the equation

For each i# 1 , 2 in N, we have a uniquely determined table ab and for does ia = 2. Also, a is one-one, since if

where i# i‘, then (ai,ai,)(p2, a j )= (ai,ai,)which is clearly impossible in 9.

20

i

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154

We note that if ia = i, then [a;,p2, pl] = a, and so ai is commutative. Furthermore, since {a3,u4,.. . , a k }is a set of m.o.l.s., at most one of the ai is commutative. We now split the proof into two parts. Case 1. No a; is commutative. In this case, the range of a is {3,4, . . . , k} and so a is a bijection. From (7.1),

ai(y,a,(x. y ) ) = a,(x, y) for all x, y in N , i.e., q ( x , u,(x, x)) = ai(x, x). But ai is a quasigroup operation and so a&, x ) = x. Hence each ai in N is idempotent (since al, a2 obviously are). Case 2. Exactly one ai is commutative. In this case, the range of a consists of N - ( 2 , h } where h is some element in { 3 , 4 , . . . , k}, since a maps the commutative ai onto ai. By the same argument as in Case 1, using (7.1) and (7.2) we know that all a, other than ah are idempotent. It remains to prove that ah is idempotent. Choose a,# ah, v E {3,4, . . . , k). W e can do this since k 2 4. Note that a, is idempotent. Then there is an a, such that

in 3. Now a,# pI and since a,# ah, a. is not equal to p . Furthermore a,# ah since a,# p2. Hence, a, is idempotent. From (7.3),

Hence, ah is idempotent and this concludes the proof.

If d = { a l ,a2,. . . uk} is an augmented set of m.o.1.s. of order n with stabilizer of order k ( k - l), then (i) k is a prime power, (ii) .d determines u Steiner (2, k)-system of order n, the blocks being the distinct sets B,j = {al(i,j ) , a2(i,j ) , . . . , ak(i,j ) } .

Theorem 7.3.

Proof. (i) is immediate since the stabilizer is a sharply doubly transitive permutation group on a set containing k objects. To prove (ii), we first note that each Bij contains k distinct elements since if a,(i, j) = as(i,j ) = f , say, then would occur in both the superimposing the two latin squares a,, a,, (j,f) ij-cell and the f,f-cell (because of the idempotency of a, and as). This contradicts the orthogonality of a, and a,. It remains to show that every pair of elements from {1,2,. . . , n } occurs in

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155

exactly one of the distinct Bit Clearly, i, j E Bij and the proof will be concluded if we show that if Bij and Biffintersect in more than one element, then they are, in fact, identical. Consider

Assume that a,(i,j ) = au(i‘,j’), a,(i, j ) = a”(?,j‘). Let ah(& j) be any element of Bit Now, if a,# ah, then, in 9, there exist ul, a, such that (ah,a,)= (al,a,)(a, a,). That is, ah = [a!,a, a,] and

This belongs to €47. Hence Bij C Eli,/. Similarly, Bin(C Bit This concludes the proof. Any set of m.o.1.s. d = {al, a2, . . . , ak} on N = { 1,2, . . . , n } may be regarded as an algebra d = (N: where is the set of binary operations al, a2,. . . , ak. We will study such algebras in more detail in the next section but one result about them occurs naturally at this stage.

a)

a

Theorem 7.4. Let d = { a , ,a2,. . . , ak}, k 2 4,be an augmented set of m.o.1.s. on N with stabilizer of order k ( k - 1). Then every 2-generator subalgebra of d = ( N :a ) is of order k , in fact, the 2-generator subalgebras are precisely the blocks of the Steiner (2, k)-system associated with the set of m.o.1.s.

hf. Let i f j be two elements of N. Then the subalgebra of .d = (N: 0) generated by i, j contains al(i7j ) , a2(i,j ) , . . . , ak(i,j ) , that is, contains the block BiP However, this set of elements is closed under the operations ai since (i) if u,(i, j ) = ar(i,j ) , then a,(a,(i,j), a,(i,j ) ) = a,(i, j) by the idempotence of the u,, and (ii) if as(i,j ) # ur(i,j ) , then u,(a,(i, j ) , a,(( j ) ) = [ a ! ,a,, ar](i,j ) . But [ a l ,a,, a,] belongs to the set of ai since the set of all (a,, at), s f t, is closed under multiplication. Note that this requires that k 4. 8. Stabilizers and Ganter-Werner algebras In [9] Ganter and Werner generalized the algebraic version of Steiner triple systems (the variety of quasigroups satisfying x2 = x, xy = yx, x x y = y ) . They considered a variety y of algebras such that if we regard the t-generated subalgebras of a I/-algebra as blocks, then the carrier of the algebra is a

156

T. Eoans. M. Francel

Steiner ( I , k)-system. However, this approach works only for t = 2, k is a prime power and for t = 3. k = 4 (apart from trivial cases). For t = 3, k = 4, we obtain the well-known algebraic version of Steiner quadruple systems. For t = 2. k = a prime power, Ganter and Werner give the identities for the 'coordin.atizing' variety V. Every algebra in V is a Steiner (2, k)-system and conversely, every Steiner (2, k)-system is represented by an algebra in We will show in this section that if d = { a , ,az, , . . , a k } , k 3 4 , is an augmented set of m.o.1.s. on N = ( 1 . 2 , . . . , n } , with its stabilizer of order k ( k - l), then regarding d as an algebra on N with binary operations a,, (some of) t h e identities which .d satisfies define a variety which coordinatizes Steiner (2, k)systems in the sense of Ganter and Werner, that is, each algebra in the variety is a Steiner (2. k)-system if we take the 2-generator subalgebras as blocks and conversely, every Steiner (2, k)-system is represented in this way.

v.

Theorem 8.1. Let d = { a , ,a2,. . . , a k } ,k 3 4, be an augmented set of m.o.1.s. on N = ( 1 . 2 . . . . , n } such that stab d has order k ( k - I). Then the algebra d = ( N : fJ) satisfies the following identities, for all x, y in N. (i) aj(x.x ) = x, i = 1,2,. . . , k , (ii) al(w(x,v). u,(x. y ) ) = a,(x, y), a,(a,(x,y ) , a , , k y ) ) = a , ( x , y ) , for every prc'duct (a,, a, ) . (q.a,,, ) = (aT,a , ) in the stabilizer of sd.

Proof. Idempotence of the ai was shown in Theorem 7.2. Equations (ii) are an

immediate consequence of the definitions of the monoid and clone operations on a l . a?, . . . . al, and Lemma 7.1.

Remarks. (1) There are, of course, other identities satisfied by the a,, not consequences of those given in the theorem. In fact, the identities (ii) have essentially nothing t o do with t h e particular a, in the set d but depend solely on the array stabilizer of d.

(2) Some other identities satisfied by the a, are worth mentioning even though they are consequences of the identities (ii). The mapping i + i ' , where ar = [a,.p2, p,] is a permutation of order two which can have at most one fixed point since a,,(x.y ) = a,(y,x) and in a set of m.0. tables, at most one can be commutative. It follows that if k is even, i.e., a power of two, that the operations u3.a 4 . .. . , ah may be paired off, i with i' and the identities

hold. If k is a power of an odd prime, then there is one a/ which is commutative,

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Some connections between Steiner systems and sets of m.o.1.s.

and the other operations in a3,a4,. . . , ak are paired i * i’ and satisfy (8.1). (3) Of course, even the identities (i) are supertluous if k 3 4 . Let yk,k 2 3, be the variety defined by binary operations 0 = {a3,a4,. . . , ak} and the identities listed in Theorem 8.1, where, as usual, a l , a2 denote the projection operations. We will denote by 9 the sharply doubly transitive group on {1,2, . . . , k} which is the array stabilizer of d in Theorem 8.1. Theorem 8.2. In any yk-algebra ( S : 0) (i) if k = 3, a3 is a Steiner quasigroup, (ii) if k b 4, the tables of a3,a4,. . . , ak are m.o.l.s., (iii) 9 is the (array) stabilizer of { a l ,az,. . . , ak}, (iv) the 2-generator subalgebras of ( S : 0)are the sets

and are the blocks of a Steiner (2, k)-system on S.

Proof. (i) is simply the example at the end of Section 4, with the idempotency law added to the total symmetry identities (these correspond to the identities (ii) in Theorem 8.1). (ii) is implied by the identities (ii) in Theorem 8.1, (iii) is obvious and (iv) is a restatement of parts of Theorems 7.3 and 7.4. In order to complete the identification of the yk with Ganter-Werner varieties, we need to show that any Steiner (2, kksystem occurs in Yk,derived contains from a yk-algebra as in Theorem 7.3. We begin by showing that the trivial (2, k)-Steiner system consisting of one block only. Define binary operations 61, c i ~ ., . . , cik on K = {1,2, . . . , k} by iii(x, x ) = x and, for x f y , di(x, y ) = i6;

(8.2)

where 0; is the permutation in the sharply doubly transitive group 9 which maps l+x, 2+y. Lemma 8.3. (i) m e algebra X = ( K ;ii,, Liz, . . ., 6 k ) belongs to the variety Yk. (ii) The Steiner system derived from 3l contains only one block, namely (1: 2,. . . , k}. Proof. (i) This follows from the definition of ai and the properties of the sharply doubly transitive group 9. cil, iiz are the projection mappings and di I cij since if

7.Evans, M.Francel

158

then i0:: = c, j O E = d and so x, y are uniquely determined as the images of 1 , 2 under the permutation of 9which maps i + c, j d. Hence {al,(zz, . . . , iik} is an augmented set of m.o.1.s. To show that X satisfies t h e identities of yk it will be sufficient t o prove that if [a,,af,a, ] = a, in any y k-algebra, then [ ii,, ii,, ii,,] = 5,. Now [a,, a,, a,] = a, implies s = i O E by (5.2) and so for any x, y in K ,

where u = lO!& v

=

mO:$ By (8.2)

Hence, [iii, ii,, ii,] = iis and rC satisfies the defining identities of yk. (ii) The blocks are { i i , ( i , j ) , ii2(i, j ) , . . . , i i k ( i , j ) } , i.e., {16;2,20L2, . . . , &oh2}. It is now an easy matter to prove that all Steiner (2, k)-systems are represented Bz,B1,.. .) be a Steiner (2, &)-system on a set S, with in Yk. Let ( S : B1, B1,&. B3,. . . the blocks. Each Biis of size k. On each Bi construct a yk-algebra Zi isomorphic to 2.Now ykis a variety defined by binary idempotent operations and the other defining identities involve only two variables. Hence, these algebras on the blocks of the Steiner system (S: B,,B2,. . .) actually determine an algebra on S which also belongs to -j f k (by [4, Theorem 101-this result is often quoted as a piece of universal algebra folklore but first occurs, at least implicitly, in [15]).

Theorem 8.4. If ( S : B,,BZ,B3, . . .) is a Steiner (2, k)-system, then there is an algebra d = ( S : 0 ) in r/" such that the sets

are the blocks B1,B2,B3,. . . of the Steiner system.

Proof. We have already observed above that (S: 0)is a yk-algebra. If i, j E S, then they belong t o a unique block B,.But, from the constru-ction of (S: a),all the eledents al(i,j ) , az(i,j ) , . . . ,ak(i,j ) are in B, and as is shown in Theorem 7.3, they are distinct. This concludes the proof that the varieties I/k do indeed coordinatize Steiner (2, k)-systems and also the proof that the algebraic structures studied by Ganter

Some connections between Steiner systems and sets of m.o.1.s.

159

and Werner are essentially the same as (i) our augmented sets of m.o.1.s. with stabilizer as large as possible, and (ii) orthogonal arrays which have conjugate invariant subgroups as large as possible. Department of Mathematics and Computer Science Emory University Atlanta, U.S.A. Department of Mathematics University of North-Carolina Greensboro, U.S.A.

References [ I ] I.M.H. Etherington, Transposed algebras, Proc. Edinburgh Math. Soc. I1 Ser. 7 (1944) 104-121. [2] T. Evans, Homomorphisms of non-associative systems, J. London Math. Soc. 24 (1949) 254-260.

[3] T. Evans, Algebraic structures associated with latin squares and orthogonal arrays. Proc. Conf. Algebraic Aspects of Combinatorics (Utilitas Math., Winnipeg, 1975) pp. 31-52. [4] T. Evans, Universal algebra and Euler’s officer problem, Amer. Math. Monthly 86(6) (1979) -73.

[5] T. Evans, Universal algebraic aspects of combinatorics, Proc. Internat. Conf. Universal Algebra, Janos Bolyai Math. SOC. (North-Holland, Amsterdam, 1980). [6] T. Evans, Some remarks on the general theory of clones, Proc. Conf. Finite Algebra and Multiple-valued Logic (North-Holland, Amsterdam, 1982). [7] M. Francel, Conjugates of sets of mutually orthogonal latin squares, in preparation. [8] M. Francel, Self-conjugate sets of mutually orthogonal latin squares, Ph.D. Thesis, Emory University, 1981. [9] B. Ganter and H. Werner, Equational classes of Steiner systems, Algebra Universalis 5 (1975) 125-140. [lo] C. Lindner and D. Steedley, O n the number of conjugates of a quasigroup, Algebra Universalis 5 (1975) 191-1%. [ 1 I] C. Lindner and E. Mendelsohn, On the conjugates of an n2 x 4 orthogonal array, Discrete Math. 20 (1977) 123-132. [ 121 C. Lindner, R. Mullin and D. Hoffman, The spectra for the conjugate invariant subgroups of n2 x 4 orthogonal arrays, Canad. J. Math. 32(5) (1980) 1126-1 139. [ 13) A. Sade. Quasigroupes parastrophiques. Expressions et identitts, Math. Nachr. 20 (1959) 73-106. [I41 S.K. Stein, On the foundations of quasigroups, Trans. Amer. Math. Soc. 85 (1957) 22G-256. [15] S.K. Stein, Homogeneous quasigroups, Pacific J. Math. 14 (1%) 1091-1102.

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Annals of Discrete Mathematics 15 (1982) 161-170 @ North-Holland Publishing Company

INCIDENCE-GEOMETRIC ASPECTS OF FINITE ABELIAN GROUPS Ulrich FAIGLE Dedicated to N.S. Mendelsohn on his 65rh birthday, April 14, 1982

With each finite abelian group A an incidence structure PG(A) involving ‘points’ and ‘lines’ is associated. Many properties of A find an immediate geometric interpretation in PG(A).If A is not a cyclic group, A can be reconstructed from PG(A).

1. Introduction

Finite abelian groups are usually studied in terms of their algebraic structure. One of the most important results, due to Frobenius and Stickelberger, is the Fundamental Theorem of finite abelian groups: every finite abelian group is the direct sum of cyclic groups of prime power order. For a finite vector space V over GF(p), it just says that V is a direct sum of one-dimensional subspaces. Now, the treatment of vector spaces admits a geometric point of view apart from the algebraic one. So one may ask whether a geometric point of view can also be taken in the investigation of finite abelian groups. Noting that affine geometry on a vector space is the geometric counterpart of the study of the system of cosets of subspaces, Wille [13] has been led to the more general concept of a ‘Kongruenzklassengeometrie’ which, for an abelian group, provides a geometrical setting for the study of the system of cosets of subgroups. Projective geometry on a vector space, on the other hand, may be seen as the study of the system of subspaces themselves. This suggests projective geometry on an abelian group as the study of the system of subgroups. The earliest result in this direction is due to Dedekind, who observed that the lattice of subgroups of an abelian group is modular. It then was especially Baer [2] who began to thoroughly investigate the connection between the structure of an abelian group and its lattice of subgroups (see also [12]). Thereby he noticed that ‘primary’ modular lattices yield a framework for a theory of both finite abelian groups and classical projective geometry ([3]; see 161

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also [ O ] ) . This theory. however, orients itself more towards aspects of coordinatization than incidence geometric properties. It turns out that finite modular lattices are naturally isomorphic to lattices of subspaces of incidence geometries involving 'points' and 'lines' that may be thought o f as generalizations of classical incidence geometries (see [6]). So finite abelian groups can also be looked at from the point of view of projective incidence geometry . I t is this 'combinatorial' approach to finite abelian groups that we want to consider here. To make the paper somewhat self-contained, t h e results needed from [6] are repeated with proof, For standard results on abelian groups and modular lattices we refer to the standard literature (e.g. [4] or 171).

2. Preliminaries

In this section we list some properties, in particular of modular lattices, that will be needed in t h e sequel. For the statements given here without proof, we refer to the standard literature on lattice theory (e.g. [ 4 ] or [ S ] ) . The subgroups of the finite abelian group A form a lattice M ( A )where t h e join is the (not necessarily direct) sum x + y and the meet the set-theoretic intersection xy of two subgroups x and y. If x =S y, then the interval [x, y ] o f M ( A ) is isomorphic to the lattice M ( y / x ) of the factor group y / x . Furthermore, M ( A ) is modular. i.e.. for all x, y. z E M ( A ) , x s z implies (x + y ) z = x + y z (here we use t h e convention that multiplication binds stronger than addition). Every modular lattice M satisfies Dedekind's 'transposition principle': for any a. b E M. the intervals [ah, a ] and [b, a + b] are isomorphic. To an element .v of a finite modular lattice M one may assign a well-defined rank r ( x ) , namely the length of a maximal chain in [0, x ] . This rank function satisfies the modular rank equality: r ( x + y ) + r ( x y ) = r ( x ) + r ( y ) for all x, y E M .

An element x # 0 of M is (join-)ineducible if x = y + z always implies x = y o r x = z . Every element x of M, if M is finite, is t h e join of the set J ( x ) of irreducibles less than or equal to x. Note that J ( x ) is an (order) ideal in the (partially) ordered set of all irreducibles of M, i.e., if r d s are irreducibles and s E J ( x ) , then also r E J ( x ) . Moreover, M is naturally isomorphic to the collection of all sets J ( x ) ordered by containment. Finite modular lattices possess t h e 'KuroS-Ore property': if x is an element of a finite modular lattice M and if x = rl + . . . + r, = s1+ . . . + s, are two

Incidence-geometric aspecfs of finite abelian groups

163

irredundant representations of x by irreducibles, then for every ri, there is an sj such that x = rl + + ri-l+ sj + ri+l+ * . * + r,, is an irredundant representation. Thus the number or irreducibles in an irredundant representation of x depends only on x. We call this number, d ( x ) , t h e (KuroS-Ore) dimension of x. Lemma 2.1. If x, y , z are elements of a finite modular lattice such that z is irreducible with z s x + y and z 9 y, then there exists an irreducible x’ s x such thatzsx’+y.

Proof. Choose x ’ s x and y ’ s y minimal with respect to z s x ‘ + y ‘ . Then z + y’ = x ’ + y’ since otherwise z y ’ + z = ( y ‘ + z ) x ‘ + y’ contradicting the minimality of x‘. Now d ( x ‘ + y ’ ) = d ( x ‘ ) +d ( y ’ )by the minimality of x’ and y’, and d ( x ’ + y ’ ) = d ( z + y ’ ) s d ( z ) + d ( y ‘ )= 1 + d ( y ‘ ) .Since z S y’, we must have d ( x ‘ )= 1; i.e., x‘ is irreducible. 0

3. The projective geometry of a finite abelian group

In the following A will always be a finite abelian group. By a (projective) point of A we understand a cyclic subgroup of order p” for some prime p and n 2 1. P = P ( A ) denotes the set of points of A (partially) ordered by containment. A (projective) line of A is a subgroup which can be represented as the direct sum of two points. L = L(A) is the set of lines of A ordered by containment. The point s is said to be incident with the line 1, denoted s E 1, if s is a subgroup of 1. The projective (incidence) geometry PG = PG(A) of A is then the incidence structure (P, L, E ) of points and lines of A. Example 1. If A is an elementary abelian p-group, i.e., if A is the direct sum of d cyclic groups of order p, PG(A) is the classical projective incidence geometry of projective dimension d - 1 associated with the vector space GF(PId. Example 2. If A is the direct sum of d cyclic groups of order p”, the maximal points and maximal lines of PG(A) are direct summands of A. Thus the incidence structure derived from PG(A) by considering only the maximal points and lines yields a’ projective Hjelmslev space (see [lo] and also [l]).

1hJ

U. Faigle

Here two points are neighbors whenever they both are greater than some point s of PG(A). A point s of order p" contains exactly n proper subgroups, which are pairwise comparable. Hence the interval [0, s] of the lattice M(A) of subgroups of A is a chain and r ( s ) = n. In particular, s is irreducible in M(A). On the other hand, if t is an irreducible element of M(A), then t must be a cyclic p-group for some prime p by the Fundamental Theorem of finite abelian groups. Thus we may state t h e following proposition.

Proposition 3.1. The points of PG(A) are exactly the irreducible elements of M(A). Moreover, M ( A ) is cyclic in the following sense: x E M(A) is irreducible iff [O. x ] is a non-frivial chain. Consequently, the ordered set P(A) of points has a very special structure: the Hasse diagram of P(A) is a disjoint union of rooted trees whose roots are the minimal points of P(A). Consider now the subgroup s + t generated by the noncomparable points s and t. Since d ( s + I ) = 2 in M(A), s + t is a direct sum of at most 2 points. But if s + t where a point, s and t would be comparable. Proposition 3.2. For any two noncomparable points s and t of PG(A), there is a unique minimal line 1 incident with both s and t. Furthermore, 1 = s + t. The projective geometry PG(A) therefore is precisely the incidence structure derived from M ( A ) by considering the elements of dimension 1 or 2.

4. Tbe lattice of subspaces

It is convenient to identify every line of PG(A) with the ideal (with respect to P) of all points incident with it. We also identify every point of PG(A) with the chain (with respect to P) of points less than or equal to it. A subspace of PG(A) is an ideal S of P such that if s, t are noncomparable points in S, S contains all points incident with the line s + f. Thus, with the identification above, every point and every line is a subspace. For any two subspaces S and T, S v T denotes the smallest subspace containing S and T.So in particular, for s. f E P, s v t = t if s S t and s v t = s + f if s and t are noncomparable.

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Incidence-geometric aspech of finite abelian groups

Proposition 4.1. Let s, tl, . . . , t k be points of PG(A). Then s E tl v * . * v s d tl + ‘ ’ + f k .

f

iff

Proof. Since the set of points contained in the subgroup tl + * . + tk clearly constitutes a subspace, s E tl v . * v fk implies s 6 tl + * * + tk. To show the converse, we proceed by induction on k. Let x = t1+ * * * + 4-1. If s 6 x or s 6 tk, then we are done by induction. Otherwise, there is a point x’ 6 x such that s 6 x ’ + tk (Lemma 2.1). By induction, x’ E tl v v tk-I. Hence s E TIv * v tk by the definition of a subspace.

-

-

--

- -

As an immediate consequence we note the following proposition. Proposition 4.2. The lattice of subspaces of PG(A) is naturally isomorphic with the lattice M(A) of subgroups.

Our next observation shows that forming. the join of two subspaces is a ‘linear’ process as in classical projective geometry. Proposition 4.3. If S and T are two subspaces of PG(A), then S v T = { u E P : u E s v t for some s, t E S U T). Proof. We must show that the set in the statement of the proposition is a subspace. But Proposition 4.2 together with a twofold application of Lemma 2.1 says precisely that u E S v T iff there exist s, t E S U T such that u E s v t.

In the special case where S = s1 v s2 is a line and T = t is a point, Proposition 4.3 states the ‘triangle property’ of classical incidence geometry (see Fig. 1): U

t

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U. Faigle

If s,, t, u, u. s2 are points such that u E s1 v C and u E s2 v u, then there exists a point x E s1 v s2 so that u E t v x.

5. Rank and dimension Because of Proposition 4.2 the notions of rank and dimension have an apparent meaning for a subspace S of PG(A): the rank r ( S ) is the size of a maximal chain of proper subspaces of S and the dimension d ( S ) is the minimal number of points needed to generate S. We will also just write r(A) and d ( A ) for r ( P ( A ) )and d ( P ( A ) ) . By the modular rank equality we see that r ( S ) + r(T)= r(S v T) iff the subspaces S and T are disjoint. Thus, if A is represented as the direct sum of points, r(A) is the sum of the exponents of the respective orders whereas d(A)is the number of those points. We remark that in group theory the term ‘rank’ is used for what we call ‘dimension’. We prefer, however, the more geometric language. Clearly, d ( A ) < r(A) with equality iff A is trivial or A is the direct sum of elementary abelian p-groups. One easily verifies that the set R of all minimal points of P is a subspace of PG(A). Moreover, r ( R ) = d ( R ) 3 d ( P ) . It will follow from the next proposition that in fact r ( R ) = d ( R ) = d ( P ) . It is not hard t o see that R is the only subspace with this property. We therefore distinguish R by calling the incidence geometry associated with R the reduced gomefry RG(A) of PG(A). Proposition 5.1. lf x < y are elements of a finite cyclic modular lattice, then (a) d , ( y ) s d ( y ) G d ( x ) + d,(y). where d , ( y ) denotes the dimension of y with respect co [x. y ] ; (b) 4 x 1 4 Y ) .

Proof. (a) If s s y is irreducible such that s S x, then [sx, x] and [ x . s + x] are isomorphic by the transposition principle and hence both are both chains. Thus

s + x is irreducible in [x, y ] . From this observation t h e inequality immediately follows. (b) Suppose that the proposition fails and choose x and y, y of minimal rank, such that x 6 y and n = d ( x ) > d ( y ) = m . So we may write y = s + u for suitable elements s. u, s irreducible, and d ( u ) = m - 1. Similarly, x = xl + . . . + x, for suitable irreducible elements xi. Since x, + x,, i = 1, . . . , n - 1, is irreducible in [x,, y ] , we must have dxa(x)= n - 1 . Furthermore, since [x,, s + u ] and [x.s, u ] are isomorphic, [0, u ] must

Incidence-geometric aspects of finite abelian groups

167

contain an element of dimension at least n - 1 by (a), a contradiction to d ( u ) = m - 1 < n - 1 and the choice of y. 0 Interpreting Proposition 5.1 for the projective geometry of an abelian group, we obtain: If B is a subgroup of the finite abelian group A, then and

0 s d ( A )- d ( A / B ) d ( B ) 0 s d ( A )- d ( B ) d(A/B).

6. Decomposition

The projective incidence geometry PG(A) of the finite abelian group A is decomposable if there are two disjoint non-empty subspaces S and T such that S U T = P. Otherwise PG(A) is indecomposable. In order to get criteria for the decomposability of PG(A) we will further investigate the structure of lines. To this end note that if A is a non-cyclic p-group generated by { a , b}, then A is generated by both {a, a + b } and {b, a + b } . In geometric terms, this observation yields the following proposition.

Proposition 6.1. If s, t are noncomparable points of PG(A) whose orders are powers of the same prime p, then there exists a point u such that s v t = s v u = t v u. We now make the following conclusion.

Proposition 6.2. If A is a finite abelian p-group, then PG(A) is indecomposable. Proof. If not, we could find subspaces S and T yielding a decomposition. Choose s E S, t E T and u E s v t such that s v t = s v u = f v u. But then u E S implies t E S, and u E T implies s E T. Hence S n TZ 0. 0

If s and f are points whose orders are relatively prime, then the sum s + t is direct and every point incident with s v t either lies below s or below t, i.e., s v t = s U t and there can be n o point u with the property as in Proposition 6.1.

Calling two points s and t perspective if they are comparable or if there is a point u such that s v t = s v u = t v u, it is therefore apparent that perspectivity defines an equivalence relation on the points of PG(A) the equivalence classes

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of which are precisely the subspaces associated with the primary subgroups of A . Moreover, since each equivalence class is determined by its minimal points, decomposition of PG(A) is equivalent to decomposition of the reduced geometry RG(A). The following proposition gives a summary. Proposition 6.3. For the finite abelian group A the following properties are equivalent: (i) A is a p-group: (ii) PG(A) is indecomposable; (iii) RG(A) is indecomposable; (iv) A n y two points of PG(A) are perspective; (v) Any two points of R G ( A ) are perspective.

7. Morphisms and projectivities

To define morphisms. we augment the sets of points and lines by ‘basepoints’. This is convenient because it allows to regard PG(A/B) as the image of PG(A) under a morphism whenever A/B is a factor group of A (Proposition 7.1). So if P(A) is the set of points of PG(A), we add a new element ep(A)as the smallest element to P ( A ) and thus get a new ordered set P’(A). Similarly, we obtain L‘(A) by adding the new element eL(A)to the set L(A) of lines. If A and B are two finite abelian groups, a morphism f : PG(A)+ PG(B) is a pair ( f p , fL) of isotone maps so that 0) f p :PYA)-+ P’(W and fP(ep(AJ= eP(B); (ii) fL : L’(A)--+ L ’ W and f d e Y A ) )= eL(B); (iii) if s is a point and 1 a line of PG(A) such that f p ( s ) # eP(B)and fL(I)f eL(B), then s E 1 implies fp(s)E fL(I). So we have a morphism from PG(A) onto the reduced geometry RG(A) by mapping every point of PG(A) onto the minimal point below it and every line onto the line generated by its minimal points. More generally, we obtain the following proposition. Proposition 7.1. If B is a subgroup of the finite abelian group A, then there is a surjective morphism f : PG(A)+ PG(AIB).

Proof. We sketch the proof by indicating the morphism in terms of the lattice WA).

The lattice of subgroups of AIB is isomorphic to an interval [n,11 of M(A).

Incidence-geometric aspects ofjinite abelian p u p s

169

Consider the map fx : M(A)+ [x, 11 : : y I+ y + x. For s E M(A), d(s) = 1, we then define

For I E M(A), d(Z) = 2, we define

It is not hard to check that

fp

and fL have the desired properties. 5

The morphism f :PG(A)-, PG(B) is a projectioity if f-’ : PG(B)+ PG(A) exists and is a morphism. Since a projectivity maps any pair of perspective points onto a pair of perspective points, we may restrict our attention to the case where A (and hence B) is a primary group. Furthermore, using Proposition 4.3, we see that a projectivity extends in a unique way to an isomorphism f :M(A)+M(B). If we represent A and B as direct sums of points, the number of points and the exponents of their orders in both representations must therefore be the same. Now, if A is not cyclic, we can find two points s and t, both of rank 1, such that s + t is incident with exactly p + 1 points. This determines the prime p for the p-group A and, by isomorphism, for B. Consequently, we obtain the following proposition. Proposition 7.2. Let A and B be finite abelian groups such that A is a p-group and d(A) 3 2. Then A and B are group-isomorphic iff there is a projectivity between PG(A) and PG(B).

Note that the hypothesis d ( A ) > 2 in Proposition 7.2 is necessary as the example of two cyclic groups of order p” and q”, p # q, shows.

8. Concluding remarks

So far we have looked at properties of the incidence geometry PG(A) derived from a given finite abelian group A. It would be interesting to be able to characterize incidence structures involving ordered sets of ‘points’ and ‘lines’ which can be derived from finite abelian groups. Those incidence structures, if they are indecomposable, must be ‘p-primary geometries’ in the sense of [6].

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But this condition is not sufficient. Even for the special case of a p-group A all of whose maximal points have the same rank, a characterization of PG(A) is not known. We point out, however, that if d ( A ) 2 4, i.e., if the representation theory of J6nsson and Monk [91 applies, a characterization is possible in terms of the characteristic of the lattice of subspaces. Ribeiro [ l l ] has also looked at abelian p-groups all of whose maximal points have the same rank and has attempted a characterization of their lattices of subgroups. As Herrmann [8] has observed, the conditions given there are, although necessary, not sufficient. This may be verified with the example of the non-isomorphic lattices of submodules M(Z :) and M(P :[x]/x'). FB Mathematik TH Darmstadt Darmstadt, W-Germany

References [I] B. Artmann. Geometric aspects of primary lattices, Pacific J. Math. 43 (1972) 15-25. [2] R. Baer, The significance of the system of subgroups for the structure of the group, Amer. J. Math. 61 (1939) 1 4 . (31 R. Baer, A unified theory of projective spaces and finite abelian groups. Trans. Amer. Math. SOC. 52 (1942) 283-343. [4] G . Birkhoff, Lattice Theory (American Mathematical Society, 3rd ed., Providence, RI, 1967). [S] P. Crawley and R.P. Dilworth. Algebraic Theory of Lattices (Prentice-Hall, Englewood Cliffs, NJ, 1973). [6] U. Faigle and C. Herrmann, Projective geometry on partially ordered sets. to appear. [7] L. Fuchs. Abelian Groups (Pergamon Press, Oxford, 1W). [S] C. Herrmann, private communication. 191 B. J6nsson and G. Monk, Representation of primary arguesian lattices, Pacific J. Math. 30 (1W)95-139. [lo} H.-H. Luck. Projektive Hjelmslevraume, J. Reine Angew, Math. 243 (1970) 121-158. I l l ] H. Ribeiro, 'Lattices' des groupes ab&ens finis. Comment. Math. Helv. 23 (1949) 1-17. [I21 M. Suzuki. Structure of a group and t h e structure of its lattice of subgroups, Ergebnisse Band 10 (Springer, Berlin, 1%7). [ 131 R. Wille. Kongruenzklassengeometrien, Lect. Notes 113 (Springer. Berlin, 1970).

Annals of Discrete Mathematics 15 (1982)171-17 @ North-Holland Publishing Company

TWO REMARKS ON THE MENDELSOHN-DULMAGE THEOREM

David GALE and A.J. HOFFMAN Dedicated to N.S. Mendekohn on the occasion of his 65th birthday We present two new proofs of the Mendelsohn-Dulmage theorem [2], one short, constructive and combinatorial, the other long and indirect and embedded in the theory of linear inequalities.

1. Introduction

The Mendelsohn-Dulmage theorem [2] plays a central role in transversal theory (cf. [3,5], for which references we are indebted to Louis Weinberg), indeed it is featured in Schrijver’s axiomatization of ‘linking systems’ [5]. The theorem originally given for finite graphs was extended to infinite graphs by Ore [4] who observed that the celebrated Schroeder-Bernstein theorem was a special case. However the proof in [4] appears to contain an error [4, p. 114, lines 14-15]. The sketch of a proof in [5, p. 261 is correct, but filling in the details is not quite trivial. The proofs of [l, 2, 31 are complete but somewhat longer than our first proof (also [l, 21 apply only to the finite case). The purpose of our first proof is to complete the argument of [5] in an effort to combine economy with completeness. Our second proof is a generalization (of the finite case) whose point is that the Mendelsohn-Dulmage theorem can be viewed as a result in the theory of linear inequalities. To state the theorem, let G be a bipartite graph with S,, S2 the partition of V ( G ) .A matching in G is a subset E C (G) such that each v E V ( G )is an end of at most one edge e E E.

Theorem 1 (Mendelsohn-Dulmage) [2]. If El and E2 are matchings in G, there is a matching E3 covering all nodes of SI covered by El and all nodes of S2 covered by E2.

2. First proof of Theorem 1 Call a node of SIa root if it is covered by El but not by E2. Consider the 171

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I>. Gale. A.J. Hoffman

components of t h e subgraph G' of G covered by E l U E2, and let Q be the set of all edges of components which contain a root. Define

(in words, choose edges of E l for components with a root, of E2 for the rest). Then E3 is a matching since its El and E2 edges are from distinct components, hence not adjacent. Further every node of G' covered by both El and E2 is covered by E3. If u in SI is covered only by El it is a root, hence covered by E3. If u in S, is covered only by E2 it cannot be in a component with a root for then there would be a path P from u to some root u, but u is in S1 hence P would have to have an odd number of edges (since G is bipartite) and the last edge of P would be in E2 contradicting the definition of a root.' Thus K is also covered by E,. We remark that for the finite case our proof is constructive. Specifically, suppose that the sets SI and S, index the rows and columns of the adjacency matrix M of G, and that the cells of M are labeled El, E2. both or neither in accordance with the given matchings. Then a root corresponds to a row containing an E , but no E2. The procedure is to look for such a row. If there is none, then let E3= E2 and stop. If there is, then change the El in this row to an E3 and repeat the procedure on the sub matrix obtained by deleting the row and column containing the label which has just been changed (this algorithm should be compared with the somewhat more complicated procedure given in [ 1,2]).

3. Second proof of Theorem 1; Mendelsohn-Dulmage matrix pairs

We begin by reformulating Theorem 1 as follows. A subset E C E ( G ) is Sk-admissible, k = I , 2, if each node of Sk is an end of at most one edge of E. Thus a matching is a subset E which is both SI-and S2-admissible. It is easy to see that Theorem 1 can be reformulated as follows. If El is SI-admissible and E2 is S2-admissible, there exists a matching E3 covering all nodes of S2 covered by El and all nodes of Sl covered by E 2 .

(3.1)

We propose to give another proof of (a generalization of) (3.1), to record that (3.1) can be viewed as a theorem about systems of inequalities. In what 'This is non-trivial detail mentioned earlier which is missing in both [4] and [ 5 ) .

Two remarks on the Mendelsohn-Dulmage theorem

173

follows, all matrices and vectors are real, A' and A' are matrices with n columns, the rows of A k are indexed by a set Sk, k = 1,2, S' n S2= 0, b' E R S 1 , b2E R*, c S d, c, d E R". Also,

P1= P(A',b', c, d ) = { x I A'x d b', c s x s d } , P 2 = P(A2,b2,c, d ) = { x I AZx b2,c s x

S

d},

T l ( x )= {i I ( A ' x )2~b:} C S1, T2(x)= {i I ( A 2 ~3) ib f )C SZ. Two such matrices A' and A' will be called a Mendelsohn-Dulmage pair if, for all b', b2,c d d, if X I E P', x 2 E Pz,there is an x 3 E P1f l P2 such that Tl(xz)u T ~ ( Xc' )T1(x3) u T2(x3). (Note that, since x 3 E P1n P2, it follows that

Tk(x3)= { i I ( A k ~ 3=)bi ! } , k = 1,2.) Theorem 2. The matrices A', A' are a Mendelsohn-Dulmage pair if and only if, for every j = 1 , . . . , n,

there are at most two i in sk such that A$ are nonzero; if there are two, they are of opposite sign ;

(3.3)

if there is an ik E Skwith A h nonzero for k = 1,2, then the two have the same sign.

(3.4)

In Section 4 we prove Theorem 2 and in Section 5 we deduce (3.1) from it.

4. Proof of Theorem 2

First we show that if A' and A' are a Mendelsohn-Dulmage pair, then (3.3) and (3.4) are necessary. Assume column 1 violates at least one of (3.3) and (3.4). Set all cj = dj = 0 for j > 1. < 0. If (3.4) is violated, then we have ii E S1 and i2E S2 so that A:,' > 0, Set bf,= Ailll,all other b: very large. Set b%= -A!2l, all other b: very large. Set c l = - 1 , d l = + l . Let x ' = (-l,O,O,. . . , O ) , x2= ( 1 , 0 , 0 , . . . , O ) . We have a

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D. Gale. A.J. H o b a n

violation of (3.2). since i2 E T2(x'),i l E Tl(x2),XIE PI, x2 E P2, and there is no possible x3. Next, assume (3.3) violated, so we have, say A:ll> 0, A:2l> 0 (the case where they are both negative is handled similarly). Let all bf be very large, let bl, = A:,, bl,= 0 , all other b! very large. Set cl = 0, d, = 1. Then X I = (0,. . . , 0 ) E PI, x2 = (1,. . . , 0 ) E P2, Tl(x2)= {il, iz} but there is no x3 E PI such that Tl(x3) contains i l and i2. Now we turn to the sufficiency. Assume x ' E PI, x 2 E P2. TI= T,(x2)CSI, Tz=- Tz(x')C Sz. We seek an x such that CAf,x,=bf, i E T l . i

zAf,xlSbf. iESI-Tl. I

i i

A;xl s bf , i E S2 - T 2 ,

c , ~ x , s d , , j = 1 . ..., n.

(4.3)

Before exploiting Farkas' theorem to test the consistency of (4.1)-(4.3), we introduce some notation. If a is a number. a

a + = (0

a ifasO, a-=(o i f a a 0 .

ifasO, ifasO,

If x = {XI, . . , , x,) is a vector, X+ = ((XI)+,. . . , (Xn)+)y X- = ((XI)-, . . . , (xn)-). By Farkas' theorem (4.1H4.3)are consistent if and only if yl E RSl, y 2 E R$, and y f a 0 foriESl-Tl,

y f 2 O foriES2-T2

(4.4)

imply (bl, y ' ) + (b'. y') -

C c,(y"A1 + y'A2)j+ - C dj(yl'A1+ y'A2)ji

We show that (4.5)holds by showing

i

30

. (4.5)

Two remarks on the Mendelsohn-Dulmage theorem

175

(y"A'+ y$A2)-= (yYA' + yrA2)-+ (yTA' + yt'A2)-.

(4.9)

Since y 1 = y : + y!., y 2 = y: + y ? , adding (4.6) and (4.7), in view of (4.8) and (4.9), will verify (4.5). We first show (4.8) and (4.9). By (3.3) and (3.4), the jth coordinate of y"A'+y;A2is the sum of at most two terms, and the only possibility of a violation of (4.8) or (4.9) occurs if it is the sum of exactly two terms and they are of opposite sign, so that

(y!A' + Y:A')~(~:'A' + Y ~ ' A '<- 0) ~.

(4.10)

But if one of the two nonzeros in column j is in A', and the other is in A', then (4.10) shows they are both multiplied by a negative coordinate of y or both multiplied by a positive coordinate of y . From (3.4) this cannot occur. If they are both in A' or both in A', the corresponding coordinates of y are of opposite sign, as are the nonzero entries in A' or A2 by (3.3), and again this cannot occur. So (4.8) and (4.9) are true. We now verify that (4.4) implies (4.6). The same argument establishes (4.7). Consider the system of inequalities i

i

A;xj c b: , i E Sz , cj d xj

S

d,

(4.11)

A:xj 2 b f , i E Tl ,

The system of inequalities is consistent if and only if 220,

imply

b:zi + iE!+

(4.12)

W S O

x b fwi - 2 cj(z'A2+ w'A')j+- 2 di(~'A'+w'A')j-

iE TI

i

20

.

(4.13)

But (4.11) is consistent, since x 2 satisfies the inequalities. Therefore (4.12) implies (4.13). By (4.4) the negative coordinates of y' are contained in TI,so

D. Gale, A.J. Hoffman

176

y!, restricted to T I ,is a nonpositive vector w, and for any vector r E RS1,whose restriction to TI we denote by I, ( w , I) = (y!, r). Set z = y : . Thus (4.13) become (4.6).

5. proof of (3.1)

Define the (0. 1) matrix M = ( m , ) with me = 1 if and only if, in the bipartite graph G, u, E S1is adjacent to u, E S2. Next, consider the system of inequalities

C ~ , ~ kj El T, 2 C S 2 . I

If E , is S1-admissible, then x = (x;,) =

1 if(vi,vj)EEl, 0 otherwise

satisfies (5.1). Similarly, 0 s x;, s nijj for all j ,

h a s a solution. derived from E2. It follows from Theorem 2 that 0 s XIj zs m;, , zxi,=1,

iET,,

~ x i j ~ ilE,S I - T l ,

jET2,

Z x j j s l , jES2-T2

I

Cxij=l. I

i

(5.3)

I

has a solution. To show there is a matching (i.e., a (0, 1) matrix x

= (xij)

Two remarks on the Mendelsohn-Dulmage theorem

177

satisfying (5.3)) it is sufficient to observe that the conditions (5.3) are totally unimodular. University of California Berkeley, USA Mathematical Sciences Department IBM Thomas J. Watson Research Center Yorktown Heights, NY, USA

References [I] E.L. Lawler, Combinatorial Optimization; Networks and Matroids (Holt, Rinehart and Winston, New York, 1976). [2] N.S. Mendelsohn and A.L. Dulmage, Some generalizations of the problem of district representatives, Canad. J. Math. 10 (1958) 230-241. [3] L. Mirsky, Transversal Theory (Academic Press, New York, 1971). [4] 0. Ore, Theory of Graphs (American Mathematical Society, Providence, RI,1%2). [5] A. Schrijver, Matroids and linking systems, Mathematical Centre Tracks 88, Mathernatisch Centrum. Amsterdam, 1978.

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Annals of Discrete Mathematics 15 (1982)179-187 @ North-Holland Publishing Company

B. GANTER and R. QUACKENBUSH* To Nathan S. Mendelsohn on his 65th birthday The classical direct method of construction of magic squares of Euler starts with a pair of doubly diagonal orthogonal latin squares. On the other hand, a magic square corresponds to a pair of orthogonal squares in a well known manner; these need not be latin squares as the magic square of Diirer shows. Interpreting the orthogonal squares from Diirer’s magic square as binary operations, we get an algebra which we call a Diiroid. Using standard techniques from universal algebra and the algebraic theory of orthogonal squares developed by Evans, we show how to define a magic square on each finite algebra in the quasivariety generated by the original Diiroid. Each non-trivial finite algebra in this quasivariety has cardinality a multiple of 4; the resulting magic squares are closely related to the well-known construction of magic squares of doubly even order.

1. Introduction

In his 1514 woodcut, “Melancholy”, Albrecht Diirer included the magic square reproduced in Fig. 1. If we subtract 1 from each cell and then write the entries in base 4, we get the square in Fig. 2. Now we form two squares, one consisting of the 4’s digits and the other of the unit’s digits (Figs. 3 and 4, respectively). We take these latter two squares to be the multiplication tables of two binary operations on (0, 1, 2, 3) (respectively denoted by and by x). This approach is quite classical; it is the approach used by Euler when he constructed magic squares from pairs of orthogonal latin squares. Our point of departure from tradition is to consider Durer’s magic square as an algebra and to apply some standard techniques from combinatorial universal algebra. We let D = ((0, 1,2,3}; 0, x): HSP(D) is the variety (equational class) generated by D. An algebra in HSP(D) will be called a Diiroid and D itself is the originalDiiroid. This paper is concerned with the algebraic and combinatorial properties of Duroids. Algebraically, Duroids are distorted vector spaces (hopefully, not distorted beyond recognition). Combinatorially, we are interested in determining when a Duroid can be interpreted as a magic square. One of the joys of combinatorics is that one can use any trick whatsoever in order to produce an example. Unfortunately, this often means that an area of combinatorics appears 0

* The work of this author was supported by a grant from the NSERC. 179

B. Canter, R. Quackenbush

180

3

2

13

33

02

01

30

5 i 10

11

8

10

21

22

13

16 I

I

I

1 9 1

6

7

12

20

11

12

23

4

15

14

1

03

32

31

00

I

Fig.3.

Fig. 4.

to be nothing more than a collection of tricks. This is certainly the case with magic squares; there is a vast literature o n magic squares, almost none in recent research journals. Our use of universal algebraic techniques forces us t o follow some rather strict ground rules. But since our approach works nicely, we gain insights which show us that the theory of magic squares is not just a collection of tricks.

2. Diiroids as algebras

In order to d o theoretical computations with the operations and X we should represent these operations in a way that makes the computations simple. An obvious thing to try is t o relate and x to a Celement group. We have two choices, t h e cyclic group of order 4 and the Klein Cgroup. It turns out that and X are closely related to the Klein 4-group. Let K = ((0, a, b, a + b}; +) be the Klein Cgroup; recall that K is equivalent to the 2-dimensional vector space over GF(2). Let a ( x ) be the autornorphism of K which interchanges a and b and fixes 0 and a + b. In the multiplication tables for and X, let us replace 1 by a, 2 by b and 3 by a + b. Then we get the multiplication tables given in Figs. 5 and 6. 0

0

0

0

Diiroids

a+b

X

0

a

b ! a + b

a+b

0

a+b

b

a

0

a

0

a

b

a+b

b

0

a

b

a+b

b

a

0

a a+bl

0

la+bla+b

a+b a+b

Fig. 5 .

Lemma 2.1. (i) x y = a ( x ) + y + a ( y ) + a + 6. (ii) x x y = x + a ( x ) + y + a + b. 0

In order to simplify computations, we replace the operations x y and x x y by 0

x‘=x+a+b,

xy=x+y+a(y).

Because x’ = x 0 x, xy = y x (x 0 x), x x y = ( y x ) ’ , a ( x ) = xx, and x 0 y = ( a ( x ) y ) ’ , we are perfectly justified in doing so. As we shall see later, the algebraic structure of D is not rich enough for our purposes. We therefore introduce an additional unary operation, x * = x + a, and denote ((0, a, b, a + b); xy, y ’ , x*) by D*. We are about to investigate the classes HSP(D) and HSP(D*). These are classes of abstract algebras and it is not clear that they have anything to do with magic squares. Suppose that A is an n-element algebra in HSP(D); we may suppose that A is based on the set (0, 1, . . . , n - 1). Corresponding to and x we construct two multiplication tables with entries from (0, 1,. . . , n - 1). Interpreting these as n’s and unit’s digits mod n, we construct a square table of side n with entries from (0, 1, . . . , n 2 - 1). Why should this be a magic square ? (By a magic square of order n we mean a square table S of side n such that each integer in {0,1, . . . , n2- 1) appears exactly once in the table and such that the sums of the entries in each row, each column and each of the two diagonals are equal.) Notice that the squares in Figs. 3 and 4 are orthogonal (each ordered pair occurs exactly once). Using the algebraic theory of orthogonal squares developed by Evans [2] (we assume that the reader is familiar with this paper), we will show that the squares corresponding to and X are orthogonal. This means that each integer from {0,1,. . . , n2- 1) occurs exactly once in S. What about row and column sums? For this, let us recall why Euler’s 0

0

B. Ganrer. R.

187,

Quackenbush

construction worked. Notice that if the squares corresponding to 0 and x have constant row. column and diagonal sum, then so does S. Euler used orthogonal latin squares of order n ; from this h e had to construct an assignment o f the entries to the integers {O, 1 , . . . n - I} so that the rows, columns and diagonals of S had constant sum. Because each symbol occurs exactly once in each row and column. every bijection to (0, I. . . . , n - 1) yields an assignment with constant row and column sum. Euler was clever enough to choose his latin squares so that an obvious bijection also caused the diagonals of S to have constant sum. O u r squares are not latin, so we cannot expect every bijection to yield a magic square. However, we will show that in our algebras there is a simple algebraic rule which tells us how to define an assignment yielding a magic square. For this we use two ideas. The first is to interpret ’ on (0, 1, . . . . n - I} by i’ = n - i - 1. The entries in the main diagonal of 0 are then of the form x o x while those of the skew diagonal are of t h e form x o ( x ’ ) . Thus ‘ permits us to get an algebraic grip o n the skew diagonal. To guarantee that we have constant row, column and diagonal sums, we insist that in any row, column or diagonal, i occurs as many times as i ’ . The main diagonal and the skew diagonal will be permutations. so we need only worry about row and column sums. We will use * to give us an algebraic grip on the pairing off of i and i f .For instance, for rows of owe will have: if x o y = z , then x o ( y * ) = 2’. Using the affine representations of xy, x’ and x * , it is easy to describe the n-ary polynomials (term functions) over D* (and also over D). We denote by FD(n)(FD.(n))the n-generated free algebra over D ( D * ) .

.

Lemma 2.2. The unary polynoniials over D* are x , xx, x ’ x , x * , x*x, x ‘ * x ; they can he represented by x. x + a + 6, a ( x ) , a ( x ) + a + b, x + a, x + b, a ( x ) + a, a ( x )+ 6, respectioely. In particular, the free algebras F D ( ~and ) F’-(l) have 4 and 8 elements, respectively. X I ,

X I * ,

Proof. Short computations show that all these polynomials are distinct and closed under composition. Lemma 2.3. Every polynomial over D* depending on the distinct variables x. x I . .. . , x . and having x as the leftmost Occurrence of a variable can be

represented by

u(x)+ where

U(X)

2 i= I

(Xi

+ a(x;))

is unary.

Proof. This is an immediate consequence of the representation of the fundamental operations by affine transformations.

Diiroids

183

Theorem 2.4. Every polynomial over D* depending on the variables x, xl, . . . , x, and having x as the leftmost occum'ng variable can be written in the following normal form:

Two polynomials are equal if their normal forms are equal.

Proof. Again this is immediate from the affine representation and from Lemma 2.3. Corollary 2.5. IFD(n)l= n2"+' and IFDa(n)l= n2"+'. Analyzing the representations of the polynomials a bit more, we find the following lemma. Lemma 2.6. A set of n polynomials in xl, . . . , x, is a free generating set of FD.(n) (or of FD(n))iff all initial variables are distinct. Corollary 2.7. IAut(FD(n))l= (n!)2"'+"and IAut(FD.(n))l = (n!)2n2+2n Next we list some identities which can be verified in D*; they are not independent. Lemma 2.8. In D* the following identities hold: (i) x ( y z ) = xy , (ii) ( X Y X Z W )= (XY ) z , (iii) (xy)y = x , (iv) ( X Y ) z = ( X Z ) Y (v) x"= x , (vi) (xy)' = x'y , (vii) xy' = xy , (viii) (xy)* = x*y , (ix) xy * = x'y , (x) x'* = x * ' . 7

Theorem 2.9. The identities (i)-(vii) of Lemma 2.8 form an equational base for HSP(D) and the identities (i)-(x) an equational base for HSP(D*).

Proof. It suffices to show that with the help of these identities each polynomial can be transformed into the normal form of Theorem 2.4. Identities (i) and (ii) can be used t o move all parentheses t o the left, (iii) and (iv) order the variables,

184

B. Canter. R. Quackenbush

and (v)-(x) can be used to shift all the occurrences of ' and * to the leftmost variable. Corollary 2.10. HSP(D) and H S q D * ) each contain the class of all sets as an equational subclass.

Proof. If we interpret xy, x' and x * trivially by xy all identities of Lemma 2.8 are satisfied.

= x,

x' = x and x * = x, then

3. Diimids as magic squares

We now wish to determine when we can interpret a Duroid as a magic square. Let A be a Duroid. If A = (0, 1,. . . , n - l } , then there is an obvious way in which A induces a square array with entries from {0,1,. . . , n 2 - 1): for 0 =si, j 6 n - 1 , the entry in row i + 1 and column j + 1 is the integer ( i 0j)n + i x j , which we shall denote by aL,.When is this square magic ? More generally, we ask whether there is a bijection L : A-+{O,1,. , . , n - 1) which induces a magic square in the prescribed manner. Such a bijection will be called an assignment; in this case, akj= L ( i j ) n + ~ ( xi j ) . We now re-emphasize that we are talking about classical magic squares: each integer between 0 and n 2 - 1 appears exactly once in the square and all rows, columns and the two diagonals have the same sum S. The diagonal from the upper left to the lower right will be referred to as the main diagonal, while the diagonal from the lower left to the upper right will be referred to as the skew diagonal. 0

Lemma3.1. { a , , , ~ O s i . j = s n - l } = { O ,.l.,. , n Z - 1 ) . Proof. The assertion of the lemma is equivalent to saying that the polynomials x 0 y and x x y are orthogonal (i.e., given x 0 y and x x y, we can solve uniquely for x and y). In universal algebraic terms this is equivalent t o saying that {x 0 y, x x y} is a free generating set for &(2) (or for FD(2)). But this follows

from Lemma 2.6.

Lemma 3.2. In every Diiroid, the columns of the table of x y and the rows of x x y are permutations. Moreover, the diagonals of both tables are permutations. 0

Proof. This follows from t h e facts that, by Lemma 2.6, {y, xo y} and {x, x x y} are generating sets of FD.(2) (and of FD(2)) and that x o x and x x x are both generators of &(1) (and of FD(1)).

Diiroids

185

Let S,,= &(n2- 1); this is the common row, column and diagonal sum of any magic square of order n. Corollary 3.3. Xiaii = S,, for every Diiroid.

From Corollary 2.10 it is clear that not every Duroid induces a magic square; if, e.g., the rows of x 0 y are constant, no L can induce a magic square for n > 1. This is a disappointing but not fatal development; we will merely try to make do without homomorphic images. Thus, let us consider SP(D), the quasivariety generated by 0,and SP(D*),the quasivariety generated by D*. We remind the reader that F’(n) E SP(D) and FD*(n)E SP(D*) for every n. In every non-trivial algebra of this quasivariety, x + x’ is an involution with no fixed points. In this case we can choose the bijection L in such a way that L ( x ) + ~ ( x ’ =) n - 1 for all x. In other words, it is possible t o arrange the elements of A in such a way that the skew diagonal elements of x o y and x x y just contain the elements of x o ( x ’ ) and x x (x’), respectively. By Lemma 2.6, these polynomials are permutations and thus we have the following lemma. Lemma 3.4. If ~ ( x+) ~ ( x ’=) n - 1 for all x, then X iai,,-i = S,,.

The only conditions remaining in this case for A t o induce a magic square are that the row sums of x o y and the column sums of x x y all equal S,,. Unfortunately, this cannot always be arranged. Lemma 3.5. F’(1) does not induce a magic square. Proof. FD(l)satisfies x 0 y

= x‘

for all x and y, i.e., x y has constant rows. 0

Thus we restrict our attention t o SP(D*), and this time with considerably more success. Notice that in D* the following identities hold: x o ( y ) * = ( x y)’ and ( x * ) x y = (x x y)’. Thus, these identities hold in SP(D*) also. The first identity implies that whenever an element z occurs in the table of x o y , then z’ occurs in the same row with the same multiplicity as z. The second identity implies the analogue for the columns of x x y. Thus if L is as above we have the following theorem. 0

Theorem 3.6. If A E SF‘(D*) and L : A + (0, 1, . . . , n - 1) is a bijection with ~ ( x+) ~ ( x ’ = ) n - 1 for all x, then A induces a magic square via L. Proof. In view of Lemmas 3.1-3.4 we have only t o show that each row of L ( X 0 y) and each column of L ( x x y) sums up to (z). But this follows from the

B. Canter, R. Quackenhush

1%

fact that L ( X ) + ~ ( x '=) n - 1 for each x E A and that both x and x' occur the same number of times in each row of x o y and column of x x y for each x E A.

Corollary 3.7. Every Diiroid. A , in SP(D*) with 2n elements induces at least ( n !)2" magic squares. Proof. An assignment L can be defined by a bijection from t h e n pairs {a, a'} in A to pairs of integers { i , n - i - I} and then on each pair {a,a'} by mapping a to either i or n - i - 1.

4. The spectrum of SP(D*) Now that we know that every n-element algebra in SP(D*) induces a magic square on {(I, 1, . . . n - l}, we turn to the obvious and important question: for which finite n 2 1 is there an A E SP(D*) with [A(= n ? That is, what is the spectrum of SP(D*)? Just as HSP(D*), the variety (or equational class) generated by D*, can be described by the identities (equations) true in D*. SP(D*), the quasivariety (implicational class) generated by D* is described by the implications true in D* (we remind the reader that these implications are all universally quantified and in prenex normal form; as with identities, t h e universal quantifiers are omitted). It turns out that by examining some of the implications true in D* (and hence true in every algebra in SP(D*)) we can readily determine the spectrum of SP(D*).

.

Lemma 4.1. If A E SP(D*) is finite and non-trivial, then IAl is a multiple of 4. Proof. For p , ( x ) and pz(.r) distinct members of {x. x', x * , x ' * } , D* (and hence A) satisfies ( p , ( x )= pz(x))+(x = y ) . Since A is non-trivial. we must have I{c, c', c * , c'*}I = 4 for every c E A. But it is now clear that (A1 is a multiple of 4 since any two such 4-element subsets either coincide or are disjoint. We are left with determining just which multiples of 4 appear in the sptxtrum o f SP(D*). In fact, we will now show that the spectrum of SP(D*) consists of I and all positive multiples of 4 (the trivial 1-element algebra appears in SP(D*)by convention as t h e product of an empty family of copies of D*).

Lemma 4.2. Each A E SP(D*) satisfies (i) (YY = Y ) - , ( X Y = (ii) ( y y = y')-,(sv = x').

187

Diiroids

Proof. It is straightforward to verify each implication in D*.

Lemma 4.3. Let d = ( d , , . . . , d,,)E (D*)"; then dd = d if and only if d E (0, a f b)". Moreover, dd = d' if and only if d E {a,b)". Proof. If d E D*, then dd

=

d iff d E (0, a + b}, and dd

=

d' iff d E {a,b}.

Lemma 4.4. Let S C (0, a + b)" be such that if d E S, then d , = 0. Let T = { d + c I d E S and c = (c, c, . . . , c ) E (D*)"}.Then T is a subalgebra of (D*)". Proof. If d E T, then d E (0, a + b)" or d E { a , b)". But if d E (0, a + b}", then dd = d, while if d E {a,b)", then dd = d'. Hence by Lemma 4.2,if d, f E T, then df E {d,f} C T. Since T is closed under ' and *, T is a subalgebra of (D*)".

Theorem 4.5. The spectrum of SP(D*) consists of 1 and all positive multiples of 4. Proof. That all non-trivial algebras in SP(D*) have cardinality a multiple of 4 is Lemma 4.1.Lemma 4.4 provides the required algebras, since I TI = 41SI.

Thus we see that magic squares of order 4k exist for all k 3 1. This is a classical result; how do our magic squares compare with the classical constructions ? Let us turn to [ 11. On page 199 we read, "A magic square of the fourth order, scarcely different from Diirer's, can be constructed by writing the numbers from 1 to 16 in their natural order in rows of four, and then replacing the numbers in the diagonal cells by their complements. The same rule applies to a magic square of any doubly-even order, if we change the numbers in those cells which are crossed by the diagonals of every component block of 4 ' cells". These magic squares cannot be constructed from the operations x 0 y and x x y , but if we replace the operation x X y with the operation y ox, then these magic squares can be constructed algebraically from Diiroids. Fachbereich Mathematik Technische Hochschule Darmstadt, W-Germany

Department of Mathematics University of Manitoba Winnipeg, Canada

References [l] W.W.R. Ball and H.S.M. Coxeter. Mathematical Recreations and Essays (Univ. of Toronto Press, Cambridge, 1914). [2] T. Evans, Universal algebra and Euler's officer problem, Amer. Math. Monthly 86 (1979) 466-473.

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Annals of Discrete Mathematics 15 (1982) 189-203 @ North-Holland Publishing Company

LATIlCE POLYHEDRA II: GENERALIZATION, CONSTRUCTIONS AND EXAMPLES H. GROFLIN* and A.J. HOFFMAN Dedicated to Rof. N.S. Mendelsohn on his 65th birthday

1. Introduction

The subject of lattice polyhedra was introduced in 181, continued in a 1976 IBM Research Rept. “On Lattice Polyhedra 11: construction and examples” by the second author, and explained further in [lo] and [ 5 ] .The present paper is a substantial expansion of this report. Lattice polyhedra are a certain class of polyhedra which not only have all vertices integral but also have the property that, when objective functions are integral, dual linear programs have integer optimal solutions whenever they have optimum solutions. Accordingly, they are suitable for proving combinatorial extremal theorems. That original motivation will not be stressed here, for such corollaries are obvious. Previously, the matrices used in the description of these polyhedra were (0, 1). Now we have them (0, 1, - 1). Definitions and fundamental properties are established in Section 2. In Section 3 we give some methods for constructing these polyhedra from some simple notions and explore the question of whether each such polyhedron arises this way. In Section 4 we show that the ideas developed capture the results of [2] (but the proof is not simpler than [2]). Finally, in Sections 5 and 6, we present a collection of specific examples, both old and new, using the methods contained here. We specially call attention to Section 6, which seems t o formulate a concept of ‘node cuts’ for directed graphs.

2. Lattice polyhedra

Let L be a finite lattice with partial order s,meet A and join v .The real valued function h : L + R is submodular on L, shorter, submodular if

h ( a v b ) + h ( a A b ) s h ( a ) + h ( b ) for all a, b E L , *While on leave from ETH Ziirich, Switzerland. 189

(2.1)

H. Griipin, A.J. Hoffman

1W

supermodular if the inequality is reversed and modular if equality holds in (2.1). The sub-, super- and modular functions to be considered here will be integer valued, 0, 21-valued and 0, I-valued and usually denoted by r, f and g. The function f : L+{O, -tl} will be said t o be consecutive on L, or shorter, consecutive if and

If(b)-f(a)I s 1 for any a, b E L with a If(a)-f(b)+f(c)(Gl

(2.2)

=S b

foranya, b , c E L w i t h a < b < c .

(2.3)

(2.2) ensures that on any chain of L, f only takes either nonnegative or nonpositive values, (2.3) that if a < b < c and f ( a ) = f ( c ) = 1 (- l ) , then f ( b ) = 1 (- 1).

Theorem 2.1. Let d and e E ( Z U & ~ r~: L, + Z , and 6:L,-+{ov*1} for j = 1, . . . n. The following polyhedra have all their vertices integer: (a) P , = {x E R ” : e 6 x s d, I:fi(a)x,S r ( a )for a E L} for r submodular, aflf,’s consecutive and supermodular, and e 5 0. @) P2= {x E R ” :e x C d, Z f,(a)x, 2 r ( a )for a E L} for r supermodular, all fi’s consecutive and submodular, and e 2 0. (c) P3 = (x E R” : e S x =sd, I:f , ( a ) x , s r(a)for a E L}for r submodular and all f i ’ s consecutive and modular. Further, for any c E 2“ and i = 1,2,3, the dual to the linear programming problem “maximize cx over x E P,” has an integer optimal solution whenever it has an optimal solution.

.

Proof. A sufficient condition for the integrality of the vertices of a polyhedron P = {n E R“ :Ax S b} is that the dual to maximize cx st. x E P, c integer, has an integral optimal solution if it has an optimal solution [7]. P is then often said to be totally dual integral [2]. The way of proving that P,, i = 1,2,3, are totally dual integral is identical to that in 181, for example, for PI: Assuming that the dual to “maximize cx st. x E PI”, minimize

ry

+ du - e v ,

st. C f ; ( a ) y ,+ u, - vj = c j ,

y E R L . U, v E R “ , j

= 1, . . . , n, y

20, u 30, o z 0

(2.4)

has an optimal solution 9, ij, 6, one numbers the elements of L a l , a2,. . . ,such that a, < a, impliesj < i. Among all optimal solutions y , u, v with l y = IY, one chooses

Lattice polyhedra I1

191

one, yo, u", v", which maximizes lexicographically y ( a l ) , y(a2),. . . . Then A = {a;E L, y:i > 0) is a chain of L, for, if ah ak E A are incomparable, let O<~Smin{y",,y;,} and y ' , u', u' such that u ' = u", uf = uy+ E ( f ; . ( a j A ak)+f;.(ai v ak)-f;.(ai)-f;(ak))for all ], y'(aj*ak)= y"(aj * @ ) +& for * = A , v, y'(al)= yo(al)- E for 1 = i, k , y ' ( a )= y"(u)for all other a. By the supermodularity of the h's, y ' , u', u' are feasible and by the submodularity of r, optimal in (2.4), contradicting the lexicographic optimality of y", u", v". By the consecutivity of the h's, and [6], the matrix M = g(a):j = 1 , . . . , n, a E A}, and hence [M, I, - I ] , is totally unimodular, ensuring the existence of an optimal integer solution j j , Lz, 6 when c is integer. 0

3. Sub- and supermodular consecutive functions Although the properties of submodularity (or supermodularity) and consecutivity are easy to state, one would like to have some combinatorial interpretation of the 0, +-1 matrices whose columns are these consecutive sub- or supermodular functions h, j = 1,. . . , n. This section characterizes these functions in terms of so called upper and lower 0,l-valued functions, introduced in ~91. The function g : L -+(0, 1) is called upper if from a s b follows g ( a )< g ( b )and lower if from a =s6 follows g ( a ) > g ( b ) . (These terms replace conveniently 'nondecreasing (increasing) 0, 1-valued'.) A consecutive 0, l-valued function g is defined similarly to a consecutive 0, ?l-valued function f by (2.2), (2.3)where (2.2) is satisfied trivially. Note that if g is upper or lower, (2.3) is also fulfilled, so that any upper or lower function is consecutive. The following two lemmata give some general constructions of sub-, super- or modular functions. Lemma 3.1. Let g', i = 1, 2, be respectively upper and lower on L. Then f : L -+ (0, kl} defined by f = g' + g 2 - 1 is consecutive. f is respectively sub-, superor modular if g', i = 1, 2, are respectively sub-, super- or modular. Proof. If a S b, 1 3 g ' ( b )- g ' ( a ) 0, - 1 s g 2 ( b )- g2(a)s 0 and (2.2) holds for f . If a s b c, 1 g'(a)- g'(b)+ g ' ( c ) 2 0, i = 1, 2, and (2.3) holds. Therefore f is consecutive. The second part of the lemma is obvious. 0 For any function h on L, denote by h' the positive part of h, i.e., h'(a) = max{h(a), 0 ) for all a E L.

H. Grojlin. A.J. H o f i a n

192

Lemma 3.2. I f f is consecutive and submodular, f ' is consecutive and submodular. Proof. As f is consecutive, f' is too. Also, the consecutivity off excludes the case: f ( a )= f ( b )= 0 and f ( a A b ) = + 1 , f ( a v b ) = + I , so that from the submodularity o f f follows the submodularity off'.

0

Remark 3.3. Changing submodular to supermodular in Lemma 3.2 invalidates it in general, as is shown by the example L = {a,6, a A b, a v b}, f ( a ) = 1, f ( b )= - 1, f ( a A 6 ) = f ( a v b ) = 0. These constructions of consecutive sub-, super- and modular functions from upper and lower functions are fairly general, as is indicated by the following decomposition lemmata. At this point, we recall that a lattice L is distributive if for any a, 6, c E L ( a A h ) v c = ( a v c ) A ( b v c ) or ( a v b ) A c = ( a A c ) v ( b A c ) , and that any finite distributive lattice is isomorphic t o a family of sets closed with respect to intersection and union.

Proposition 3.4. Let f : L + (0. + 1) be consecutive and gl, g2 be defined by g'(a) = maxCf(b): b s a ) , g'(a) = maxCf(b): b 2 a ) . Then each g' is consecutive and either nonpositive or nonnegative. Also, i f f is supermodular. g' and g 2 are supermodular; if f is submodular and L is distributive, g' and g z are submodular; i f f is modular and L is distributive, g' and g 2 are modular. (i) For any a s b, g ' ( a )C g'(b), g2(a)== g2(b)and (2.3)is easily verified for both g' and g z . Further, if g ' ( a )= -1, g'(b) = 1 , let d and e be such that d S a, e s b and f ( d ) = g'(a), f ( e ) = g'(b). By consecutivity of f , d and e are not comparable, but then f ( e A d ) = 0 , contradicting g ' ( a )B f ( e A d ) . Therefore each gi is either nonpositive or nonnegative and (2.2)is satisfied; therefore g' and gz are consecutive. (ii) Assume f supermodular and let d, e be such that d S a, e S b, f ( d ) = g ' ( a ) and f ( e )= g'(b). As d A e < a A b, d v e S a v b, g'(a A b ) + g'(a v b ) a f ( d A e ) + f(d v e ) 3 f ( d ) f ( e )= g ' ( a )+ g'(b), i.e., g' is supermodular. Analogously one shows that g 2 is supermodular. (iii) Assume f submodular and L distributive. If g'(a A 6 ) = 1, g'(a v b ) = ~ ' ( a= )g ' ( b ) = 1. I f g l ( a A b)
+

Lanice polyhedra ZZ

193

A b ) = -1, by consecutivity off, g'(a v b ) c O , and therefore g'(a v 6 ) = g'(a) = g'(b) = 0 and g' is submodular. Analogously, one shows that g2 is

f(d A a

submodular. The last part of the proposition is obvious. 0

Similarly one proves the following proposition.

Proposition 3.5. Let f : L + (0, +1} be consecutive and g'(a)= minCf(b): b 5 a}, g2(a)= minCf(b): b 3 a}.

g', g2 be defined by (3.2)

Then each g' is consecutive and either nonpositive or nonnegative. Ako, i f f is submodular, 2' and S2 are submodular; i f f is supermodular and L is distributive, g' and g2 are supermodular, and i f f is modular and L is distributive, g' and g2 are modular. Lemma 3.6. Let g : L + (0, 1) be consecutive and g # 0. There exists g' and g2 upper and lower such that g = (g' + g2- 1)+ and (i) g' and g2 are supermodular if g is supermodular; (ii) g' and g2 are submodular if g is submodular and L is distributive; (iii) g' and g2 are modular if g is modular and L is distributive. Proof. Let g' and g2 be given by (3.1) for f = g. If g f 0, g' and g 2 are nonnegative and therefore respectively upper and lower. If g(a)= 1, g'(a)+ g2(a)- 1 = 1, and conversely by consecutivity of g. Thus g(a) = 0 implies (g'(a)+ g 2 ( a ) - 1)+ = 0 and conversely. The super-, sub- and modularity of g', g2 follow from Proposition 3.4. 0

Remark 3.7. (g' + g2- I)+ = g' + g 2 - 1 iff every a E L such that g ( a ) = 0 is comparable with some b E L with g(b)= 1. While this always holds for a supermodular, nonnegative, nonzero function g, it does not generally for submodular functions, as is shown by the example L = {a, b, a A b, a v b}, g ( a ) = 1, g ( b ) = g(a A 6 ) = g(a v 6 ) = 0. Lemma 3.8. Let f : L + (0, f1) be consecutive and f # 0. (i) I f f is supermodular and f ( @ ) = 1for some E L, there exists g' and g2 supermodular upper and lower such that f = g' + g2- 1 . (ii) I f f is submodular and f (a,,)= - 1 for some E L, there exists g' and g2 submodular upper and lower such that f = g' + gz- 1. (iii) I f f is modular and L is distributive, there exists g' and g2 modular upper andlowersuch t h a t f = g ' + g 2 - 1 .

194

H. Groflin, A.J. Hoffman

Proof. (i) Let f be supermodular and g' and gz given by (3.1). Clearly, g'(u,,) = 1 and therefore g' 2 0, i = 1, 2. f(a)= 1 implies g'(a)+ g'(a) - 1= 1 and conversely. Assume f(a)= 0. As f(6) = 1, if a > a(] or a
modularity of gI and g' follows from Proposition 3.4. (ii) If f is submodular, f = -f is supermodular and f ( a o )= 1. By (i) T= g' + g2- 1, where g ' ( a )= max{f(b): b S a} = -minCf(b): b S a} = -g'(a), i.e., g' = - f . i = 1, 2. Therefore f = g' + g2+ I with j ' and j 2 defined by (3.2), or f = g 1 + g 2 - 1 , with g ' = g ' + l , i = l , 2 . As g ' 6 0 , i = l , 2, g' and g 2 are submodular lower and upper. (iii) Iff is modular and there is a. E L such that f(ao)= 1 , f = g' + g2 - 1 with g' and g' given by (3.1). If f is modular and f(a0) = -1 for some ao, f = g' + g2- I with g' = g' + 1 and S' given by (3.2), i = 1, 2. The modularity of g ' and g 2 follows from Proposition 3.4 or 3.5. 0 Remark 3.9. Using Lemma 3.6 and Remark 3.7 one can complete (ii) and (iii) of Lemma 3.8 by showing that if f is submodular and f 5 0 , f = g' + g 2 - 1, g I and g2 given by (3.1), provided that L is distributive and every a with !(a) = 0 is comparable with some b with f ( 6 ) = 1, and if f is supermodular and fco, f = g ' + g 2 - 1, g' = g ' + 1, g' given by (3.2), i = 1, 2, provided that L is distributive and every a with f ( a )= 0 is comparable with some b with f(b) = -1. Note that, letting for instance f a 0 submodular, if either of the two conditions: L distributive or every a with f(a) = 0 comparable with some 6 with f ( b ) = 1 fails to hold, then there might not be any h ' , h 2 upper and lower submodular such that f = h' + h 2 - 1, as the examples of Fig. 1 show.

Fig. I . In both examples. L is given by the Hasse Diagram. The numbers are the values off on L (zero values have been omitted).

Lattice polyhedra II

195

4. Crossing and cross-free subsets

The results of this section extend Theorem 2.1 to polyhedra defined on certain subsets of the lattice and has been motivated by the aim to bring together the class of polyhedra given in [2] by Edmonds and Giles with the lattice polyhedra. Their concepts of crossing and cross-free families of sets are borrowed here. Let L be a distributive lattice with minimal and maximal elements m and M. A is called a crossing subset of L if for any a, b E A, a A b# m and a v b# M implies a A 6, a v b E A. A is called a cross-free subset of L if for any a, b E A, a A b# m and a v b # M implies that a and b are comparable.

Lemma 4.1. Given L as above and A a cross-free subset of L, let fi : L + (0, k l}, = 1, . . . , n, be consecutive and modular such that A(m) = A(M) = 0 for ail j , Then the matrix B = Cf;.(a),a E A, j = 1, . . . , n } is totally unimodular.

j

Proof. As any subset of A is cross-free, it is sufficient to assume B to be square and show det B = 0 or 21. Denote by f ( a ) the ath row of B (and more generally the vector with components f l ( a ) ,. . . , fn(a ),for any a E L) and by 0 the zero row. If A is a chain, det B = 0 or 2 1 [6]; if not, and there is a, b E A incomparable such that a A b = m, choose a maximal such pair a, b (maximal meaning: if c, d E A incomparable, c A d = m, c z= a and d 3 b, then c = a, d = b ). Adding the row f ( b ) t o the row f ( a ) yields the row f ( a v b). Let B1 be the resulting matrix with row set A, = A U { a v b} - { a } . Det B1= det B. If we show that A , is again cross-free, we can repeat the same operation with Al and B1, etc., generating a sequence B, B1,. . . , Bp of matrices with same determinant. If this sequence is finite, all pairs of incomparable elements a, b of the cross-free set Apwill be such that a v b = M. Assuming now that A, has such pairs, let a, b be a minimal such pair (where minimal is the obvious counterpart of the maximal defined above). Adding f ( b ) to f ( a ) yields f ( a A b ) and B,+, with row set A,+, = A, U { a A b } - { a } . Det Bp+l= det Bp. Again, A,+, turns out t o be cross-free. Moreover, checking all possible cases, it is easy to see that if c and a A b are incomparable, c v ( a A b ) = M, i.e., all the pairs of incomparable elements c, d of A,,, are such that c v d = M. Repeating the same operation yields a finite sequence BP+,,. . . , B, where A, is a chain and therefore det B = det Bq = 0 or 2 1 . We prove now that A, = A U { a v b } - { a } is cross-free and that the sequence Al, . . . , A, is finite. Assume det BZ 0, and c E A with a v b and c incomparable. Essentially two cases can occur: (i) a < c and b, c incomparable, (ii) a, c

H. Wjh, A.J. Hofman

1%

and b, c incomparable. In (i), by the maximality of a, b, b A c f m and b v c = M, therefore (a v b ) v c = M and ( a v b ) A c # m. In (ii), as a A b = m, a A c = b A c = m, and therefore ( a v b ) A c = m and ( a v b ) v c # M. Hence A, is cross-free. Also, if ( a v b ) A c = m, both a A c and b A c = m, therefore there are less incomparable pairs of the type a A b = m in A, than in A, ensuring the finiteness of the sequence. Similarly, one shows the validity of the sequence Ape,.. . . , A,.

0

A function h : A + R is said to be submodular on the crossing subset A if for any a, b E A such that a A b# m and a v b# M, the submodular inequality (2.1) holds.

Theorem 4.2. Let L be a distributive lattice with minimal and maximal elements m and M, f i : L + { 0 , kl} consecutive and modular with h ( m )= A(M) = 0, j = 1. . . . , n. A a crossing subset of L, r : A + Z submodular on A and d, e E {Z u fa}".Then

{

P i = x E R " : e s x s d , x f i ( a ) x j S r ( a ) a, E A ] is totally dual integral and all its vertices are integer.

Proof. Let yo, uo, vo be an optimal solution to the dual to the LP: maximize cx, c, x E R", such that x E P ;

minimize ry such that

+ du - ev,

x fi(a)yll+ uj

- vi = c ,

y E RA, u, v E R " ,

j = 1, . . . , n, y, Y v 3 0

9

and let A. = { a E A : y: > 0). Clearly, in view of the proof of Theorem 2.1, one can assume that A. is a cross-free subset of L. By Lemma 4.1, M = Cfi(a), j = 1, . . . , n, a E Ao} and therefore [M, I, - I ] is totally unimodular, ensuring the existence of an optimal integer solution 1, ii, ij when c is integer. 0 A weaker result holds for some super- and submodular functions.

Lemma 4.3. Let L be a lattice with minimal element m, A C L such that for any a, b E A: a A b # m implies that a and b are comparable, hi : A + {0,1}consecutive functions on A with h j ( m )= 0 and such that if a, b E A and a A b = m + h j ( a ) + h , ( b ) S 1 , j =1, . . . , n. T h e n t h e m a r r i x B = { h j ( a ) , a E A , j =1, . . . , n } i s totally unimodular.

Lattice polyhedra II

197

Proof. We may assume B to be square with det B# 0, and represent A as an arborescence A':letting al, . . . , a, be the maximal elements of A, represent them as edges with tails t(ai)= r, i = 1, . . . , s, r being the root of A'. For each remaining element a of A, construct an edge with its tail t ( a )= h(b),the head of b E A, if b covers a. Any column of B is then easily seen to be the edge incidence vector of a directed path in A' and B is therefore totally unimodular. 0

Using the method of proof of Theorem 2.1 and the Lemmata 4.3 and 3.2, it is straightforward to prove the following theorems, with L a n d A given as in Lemma 4.3.

Theorem 4.4. Let gi : L -,(0,1} be consecutive and supermodular with gi(m) = 0 , j = 1,. . . , n, r : A+Zsubmodular on A and d, e E{Z+U+m)n. Then

Pi = ( x E R":e s

x =sd,

2 g,(a)x,

r(a), a € A }

is totally dual integral and all its vertices are integer.

Theorem 4.5. Let f;. : L-, (0, kl} be consecutive and modular with f ; . ( m= ) 0, j = 1,. . . , n, r : A + Z supermodular on A and d, e E {Z+US=+'. Then x E R": e 5 x

Id,

2 fT(a)xj2 r(a),a E A

I

is totally dual integral and all its vertices are integer.

5. Some applications Before giving any examples, we would like to point out that, given any lattice L, there are only a finite number of possible functions : L+(O, kl}, which are sub-, super- or modular. Consequently, apart from choices of r, d and e, there is essentially only one example in each of the three classes. It is, nevertheless, interesting to look at special instances. Therefore we give here some applications of the Theorems 2.1, 4.2 and 4.5 and the construction Lemmata 3.1 and 3.2, yielding some known and new integer polyhedra. 5.1. Let L-,, be a lattice, L a sublattice, E C L-,,. Then for any j E E, gi : L +

(0, I} defined by

H. Grogin. A.J. H o b r a n

I98

1 ifjsn. g J ( a )= {(I otherwise .

(5.1)

is upper and supermodular. Let additionally LObe distributive with minimal and maximal element rn and M. then gt, i = 1.2. defined by 1 ifjAa=m. otherwise,

(0

1 ifjva=M. {0 otherwise.

are respectively lower and upper supermodular. Using (5. l), (5.2) and Lemma 3.2,

is of type Pz, for b distributive, L a sublattice of LO,E C LO and r supermodular. In case b is the lattice of all subsets of V = (0, 1, . . . , n}, L the sublattice of all subsets of V containing 0, E the two-element subsets of V, one has the cut set polyhedron [ 111 for undirected graphs. 5.2. Let L , and Lz be two lattices and g' : Li+{O, I}, i = 1, 2, be respectively upper and lower supermodular. Define L = Ll U Lz where the element set of L is the union of the element sets of Liand L2, with the ordering in L I and L2 the same as before and with a < b for a E L,, b E L 2 . f : L+ (0, l}, defined to agree with g' on L,, i = 1, 2, is consecutive and supermodular. Given now a lattice L and g upper and supermodular, and letting L be the lattice obtained by reversing the order (and consequently interchanging A and v), g : L+{0, 1) is lower and supermodular. Therefore, letting L , = L and Lz = and given upper supermodular functions g,, j = 1 , . . . n, and submodular functions rl, rz : L + 2,

c

.

[x E R:: 2 g,(a)xj

G

ri(a),a E L, i

=

1,2

(5.3)

is of the type Pz. (If the gj are upper modular, by Theorem 4.2, t h e constraints can be restricted to a crossing subset.) Using (5.3) one can obtain the 2polymatroid intersection polyhedra [ 1,2].

5.3. Let G = ( V ,E) be an undirected graph with node set V and edge set E, L be the lattice of all subsets Ei, i E I, of E, and V ( E i )the set of the endnodes of

Lattice polyhedra

I1

1%

the edges of Ei. Then gj defined by gj(Ei)= 1 if j E E i , = O Otherwise, is upper and modular, and r by r(Ei)= I V(Ei)I- lEil is submodular, therefore

[x E R

:

2 {xi: j E Ei}=s I V ( E j )-( IEi1,

i EI ]

is of the type P3 and one obtains a version of t h e connected polyhedra of [4]. Clearly, in view of (5.3), a corresponding intersection polyhedron for two graphs G' = (Vk,E ) , k = 1,2, is also of the type P3.

Li,i = 1, 2, be two lattices, g' : Li+{O, l), i = 1, 2, be respectively upper and lower sub(super)modular. Let L = LI x L2 with (a, b ) E L if a E LI, b E L2, and (a, b) S (c, d) if u S c in Ll, b =s d in L2. Clearly, g" : L-,{O, 1) such that g"(al, a2)= g i ( a j ) for all (al, a2)E L, i = 1,2, are upper and lower sub(super)modular. In particular, let L1 be the lattice of all subsets of some set E, L2 = Ll and g' : Li +{O, 1) defined by gj(S)= 1 if j E S, =O otherwise, i = 1, 2. By Lemma 3.2 J(S,T)= 1 if j E SnT, =O otherwise, is consecutive and submodular on LI X El. This is used in deriving a result on matroid intersection in [ 5 ] . 5.4. Let

5.5. Let G = (V, E) be a directed graph and denote by t ( j ) and h ( j ) the tail and the head of edge j E E. Let L be a lattice of node subsets S C V. For any j E E, g,! : L-, (0,l) defined by gj(S) = 1 if t(j) E S, =O otherwise, is upper and modular, gf defined by gf(S)= 1 if h ( j ) S, = O otherwise, is lower and modular. Using Lemma 3.1, Theorem 4.2 and J(S) = gf(S)+ gf(S)- 1= 0 for S = 0 or V, for all j E E , one obtains the polyhedra of [2], and using Theorem 4.5, a polyhedron of [3]. 6. Node cuts in directed graphs 6.1. Let G = (V, E) be a directed graph and L a lattice of node subsets S C V.For any u E V, gl : L+{O, 1) defined by g : ( S ) = 1 if v E S, =O otherwise, is upper modular, g:' defined by g':(S)= 1 if u has n o successor node in S, =O otherwise, is lower and supermodular. Using Lemma 3.1 and Theorem 2.1 and IettingfJS) = 1 if u E S and u has no successor in.S, = -1 if v $Z S and v has a successor in S, =O otherwise, d being integer and r submodular and integer, the polyhedron

is totally dual integral and has its vertices integer.

200

H. Grii@n,A.J. Hoffman

6.2. Let G‘ = (V’. E’) be the line digraph of the directed graph G = (V, E), i.e.,

V’= E and ( u ; , &)E E’ iff h ( u i ) = t ( u ; ) in G. Let L(V) be any lattice of subsets S C V and for any S E L(V), let T ( S ) C V‘ be defined by T ( S ) = {u’E V’:r(u’)E S}. Clearly, L = { T(S):S E L( V)}is a lattice of subsets of V‘ and defining for G’ the functions g’ and g” of 6.1 gives modular upper and lower functions, in fact precisely the functions g’ and g2 of 5.5 for G.

6.3. In a directed graph G, let c : V + R such that X{cu: u E C}sO for any directed circuit C of G. We shall show that the problem of finding a longest path in G (i.e., a path D such that Z{ct,: u E D ) is a maximum) has a curious linear programming formulation. I n 6.1, let L be the lattice of all subsets of V, r ( S ) = 1 for all S C V, and d = +m. Denote by f(S) the vector with components f v ( S ) , u E V. so that we consider P = { x E R Y : f ( S ) x s l , S GV). Let a’. . . . , u p and b’, . . . , bq be the node set incidence vectors of all directed paths (including the null path) and of all directed circuits of G and let N

+ Cone(b’, . . . , b9)

= Conv(a’, . . . , a”)

be the sum of the convex hull and the cone generated respectively by a’, 1 sz i d p. and b’, 1 S j s q. We will show that P = N. Clearly, this implies that max cx, such that x E P, under the assumptions on c made above, solves the longest path problem. Clearly N P, as for any ai, bi and S,f(S)ai d 1andf(S)b’ 0. Toshow P C N, we use the fact, which we will prove later, that Q = {x E R v : a’x < 1.1 S i S p, b’x d 0 , l s j

S

4)

has all its vertices z ’ , . . . . , z‘ with components 0, +1. Then, by polarity results, N

= {x E RY: x i

s 1 for all z E Q}= {x E RY: xzk S 1 , l S k

S

r} .

To any maximal z E Q with 0, k1components, one can associate an S V such that z =f(S) in the following manner: Let SO={ u E V :zu= l}, S,,, = S,,.. U { u E V ; z, = 0 and u has a successor in Sm-l},rn = 1,2, . . . . Let S be the largest such set S,,,. Clearly, if u $Z S has a successor in S, zj = - 1, if u E S is such that z, = 1, as I E Q, u has n o successor in S, and for all u $? S with no successor in S, z, = 0 by the maximality of z in Q. Therefore P CS N.

,

Lattice polyhedra I1

201

Now, for any vertex iof Q, as all singletons (or one vertex paths) are among the directed paths, f S 1, and as f is maximal in 0, f 3 -1. That Q has all its vertices integer, and therefore 0, +1, can be seen by proving total dual integrality. The dual t o “maximize cz such that x f Q” is min lu, st.

u E RP,

2 uiai+

w E Rq,

wj& = c,

u, w B

(6.1)

o.

Let V = {ul, . . .,u,} and consider the following transportation problem: given are origins Om,m = 0 , . . . , f, with capacities w, c l , c2,. . . ,ch destinations D,,, n = 0, . . . , f with demands w , cl. c2, . . . ,c,, and the cost matrix {rm,,} with rmo= 0 f o r m = 0,. . . , f, Ton = 1for n = 1 , . . . , f, r,,, = O i f (urn,u,,),)EE,=motherwise, for 1 S m, n S f ; w is chosen to be a large number. With any solution y w to (6.1), one can construct a feasible solution {&,, 0 S m, n S f } of the transportation problem by letting, for 1d m, n S t, (on = X{ui: u,, initial node of path i}. &,o = X{ui:urn terminal node of path i}, 500= w - X L I to,, = w - X ui, &,, = X{w: urnprecedes u. in path i} + X{wj:u,,, precedes u,, in circuit j } . Moreover X rmn&,n = X u, = lu. Conversely, given any solution {&,,,, 0 G m, m s t}, one can construct a solution u, w to (6.1) such that Z rrn,,~mn = lu, by using the following path or circuit generating procedure: Let initially u = 0, w = 0. Starting with k(0) = 0, one selects step by step a sequence of positive components ( k ( 0 ) . k ( 1 ) , ( k ( l ) , k @ ) , . . * (k(h-1). k(h), k ( 1 ) S 1, until k ( h ) = k ( s ) for some 0 g s < h - 1. If k ( h ) = 0, one has a path a’ with nodes uk(l), . . . , Vk(h-l)r sets u, = min{[k(,), k(s+l): 0 s s < h} and reduces Sk(s).k(s+l), 0 =S s < h, by u;. If k ( h ) = k ( s ) for 0 < s < h, one has a circuit & with nodes u k ( s ) , . . . ,t)k(h-l) and sets wj = min{[k(ssxk(s,+l): s s s’< h} and reduces (k(s’). k(s’+l) by WJThis procedure is repeated until 5O.k = 0 for 1 6 k S f. Then one proceeds with 0 < k(0)s f, generating only circuits. The application of this procedure is easily seen to reduce the solution &,,, 0 d m, n c f, to zero. Furthermore l u = X rm,,&,,,. If w and c,,, II = 1 , . . . , t, are integer, one can find an optimal integer solution 6 to the transportation problem and derive from it an integer solution t o (6.1), which by the above observations, is optimal for (6.1). 3

6.4. If G is an acyclic graph, an alternative proof that P of 6.3 is the convex hull of the node set incidence vectors a’, i = 1, . . . ,p, of the paths of G is provided by

solving the LP’s (letting L = {S: S C V)): maxcx, x E R ” , st.

c.fu(S)x,

S

1,

s E L, x 3 0 .

H. Groflin, A.J. Hoffman

202

minly,

yERL,

We solve (6.2) by determining with dynamic programming a longest path: for each u E V, calculate uc, the length of an optimal path starting at u. by setting u, = max(0, c,} for all u with no successor node, and computing recursively for all remaining nodes u : u, = max(0, c,.

+ ux,:w successor of

u}

u* = max{y,: u E V) is the optimal value of (6.2),as will be shown by exhibiting a feasible solution to (6.3) with the same objective value. An optimal path and its incidence vector are then easily obtained by ‘tracking’ from a node u , with 14(u*) =

u’.

The dual solution is obtained with the following simple algorithm:

v,

Step 1. Number the nodes such that, with n = I u13 uz2 .. . > u,. S t e p 2 F o r i = 1. . . . , n,setS,={l, . . . , i}andys,=u,-u,,l(definingu,,+l=O). Set y s = O f o r a l l S ~ { S , : i =1, .... n } . Clearly 1y = u* and y 3 0. Also, observe that for any k E { 1. . . . , n } , if k has no successor, f k ( S , )= 1 for i = k, . . . , n, and = 0 otherwise; if k has a successor, and t denotes a successor with largest U-value, depending on whether k > 1 or k < 1, f k ( S , ) = 1 for i = {k,.. . , I - l}, and =0 otherwise, or f k ( S , ) = -1 for i = { I . . . , k - 1) and =O otherwise. In all cases one verifies Z f k ( S ) y s3 ck. As (6.2) is solved by the node set incidence vector of an optimal path for any c, any vertex of P is a node set incidence vector of a path. The converse holds clearly. Also. if c is integer, the dual solution obtained above is integer. IBM T.J. Watson Research Center Yorktown Heights, NY, USA

References [ I ] J . Edmonds. Submodular functions. matroids and certain polyhedra, Comhinatorial Structures and their Applications (Gordon and Breach, New York, 1970) pp. 69-87. [2] J Edmonds and R. Giles, .4 min-max relation for submodular functions on graphs, Studies in Integer programming. Ann. Discrete Math. I (1977) 185-204. 1-71 A. Frank. Kernel systems of directed graphs, Acta Sci. Math. (Szeged) 41 (1979) 6.-76. 141 H. GroRin and Th.M. Liebling, Connected and alternating vectors. polyhedra and algorithms, Math. Programming 20 (1981) 23F244.

Lattice polyhedra II

203

[5] H. Groflin and A.J. Hoffman, On matroid intersection, Combinatorica 1 (1981) 43-47. [6] A.J. Hoffman and J.B. Kruskal, Integral boundary points of convex polyhedra, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities and related systems, Annals of Mathematics Study 38 (Princeton University Press, Princeton, 1956) pp. 223-246. [7]A.J. Hoffman, A generalization of max flow-min cut, Math. Programming 6 (1974) 352-359. [8] A.J. Hoffman and D. Schwartz, On lattice polyhedra, in: A. Hajnal and V.T. S6s, eds., Combinatorics: Proc. 5th Hung. Coll. on Comb. (North-Holland, Amsterdam, 1978) pp. 59s598. [9] A.J. Hoffman, On lattice polyhedra 11: construction and examples, IBM Research Rep. RC 6268, 1976. [lo] A.J. Hoffman, On lattice polyhedra 111: blockers and anti-blockers of lattice clutters, Math. Programming Stud. 8 (1978) 197-207. Ill] E. Johnson, O n cut set integer polyhedra, Cahiers Centre Etudes Rech. Op&. 17 (1975) 235-251.

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Annals of Discrete Mathematics 15 (1982) 205212 @ North-Holland Publishing Company

A PARTIAL GEOMETRY pg(9,8,4)

W. HAEMERS* and J.H. van LINT Dedicated to N.S. Mendelsohn on the occasion of his 65th birthday

We describe a construction of a partial geometry pg(9, 8, 4) based on binary words of length 9 and PG(1, 8).

1. Introduction A (finite) partial geometry S = (9, 53,I) is an incidence structure with a symmetric incidence relation satisfying the following axioms: (i) Each line is incident with s + 1 points (s 3 1) and two distinct lines are incident with at most one point. (ii) Each point is incident with t + 1 lines (t 2 1) and two distinct points are incident with at most one line (here the second assertion is implied by (i)). (iii) If x is a point and L a line, such that x d L, then there are exactly a (a2 1) points XI, x2,. . . , x, and a lines L1,L2,. . . , L, such that x I LiI xi I L ( i = 1, 2 , . . . , a). Although the numbers s, f and a are called the parameters of the partial geometry, we denote such a partial geometry by pg(s + 1, t + 1, a) because the numbers in brackets correspond to the objects which are counted by (i), (ii) and (iii). We call the points x and y collinear if there is a line incident with x and y. We denote this b$ x y . If S = pg(s + 1, t + 1, a) then the corresponding graph T ( S ) is defined as follows. A vertex of T ( S )is a point of S and {x, y} is an edge iff x - y . It is easy to check that T ( S ) is a strongly regular graph with parameters

-

n

= (s+

p = (f

l)(st+ a ) / a ,

+ 1)a

k

= s(t+ l),

A = (s- 1)+ t ( a - l ) ,

and that the eigenvalues of the adjacency matrix of T ( S ) are s ( t + l), s - a, multiplicity 1, f:= s t ( s + l)(t + l ) / a ( s+ f + 1- a)and n -f- 1 respectively (cf. [l, 71).

--I - 1 of

* Present address: Dr. Neher Laboratories, Leidschendam, The Netherlands. 205

W.Haemers, J.H. van Lint

24%

A partial geometry is called a proper partial geometry if 1 < a < min{s, t } . Until a few months ago the only known infinite classes of proper partial geometries were those constructed using arcs in projective planes of order 2h. All these geometries have odd a. The only other known proper partial geometry was pg(6.6.2) constructed by Van Lint and Schrijver (cf. [6]) using codes. Subsequently Cohen constructed a pg(9,8,4) using group theory (cf. [2]). In this note we describe a much simpler representation of such a geometry. These two constructions inspired De Clerck, Dye and Thas (cf. [3]) to construct a new infinite class of partial geometries ~ g ( 2 ~ "+- '1, 22"-',22n-2)which is more or less naturally embedded in projective geometries (e.g. pg(9,8,4) is associated with a quadric in PG(7.2)). The idea of this construction was generalized by Thas, producing a method for constructing ~ g ( 3 ~ "+- 1,3'"-', ' 2.32"-2)which works for at least two (and probably for all) values of n (cf. [8]). At that point (early 1980) the geometry pg(6.6,2) was once again the only 'sporadic' proper partial geometry. Quite recently (June 1980) Haemers constructed a second one, namely pg(5, 18,2) (cf. [S]).

2. Some lemmas Partial geometries were introduced by Bose [ l ] t o study large cliques in strongly regular graphs. We first give a bound on the size of a (co-)clique which is well known. For a proof using matrix techniques we refer to [4]. W e give a proof by a double counting method (which we have seen referred to as Mendelsohn's method).

Lemma 1. Let G be a strongly regular graph with parameters n, k, A, p and let u be the negative eigenvalue of the adjacency matrix of G, i.e. the negative root of the equation x' + ( p - A ) x + p - k = 0. Let C be a coclique of size c. Then,

and equality holds i f f each vertex outside C is adjacent to kc/(n - c ) vertices of

c..

Proof. Let x, be the number of vertices outside C which are adjacent to i vertices of C. Then we have

Cx,=n-c,

x i x i = kc,

A parrial geometry pg(9, 8,4 )

207

Hence,

z ( i - kc G ) x i = p c ( c - l ) + k c - - k2cZ n-c is nonnegative. This implies

p ( c - 1 ) + k - k2c/(n- c ) 2 0 . W r i t e x : = - k c / ( n - c ) . By useof k ( k - A - l ) = p ( n - k - 1 ) w e f i n d x'+(p-A)X+p-k

hence v

SO,

x and both results follow.

A strongly regular graph G with parameters n, k, A, p is called pseudogeometric if there are integers s, t, a such that a partial geometry S = pg(s + 1, t + 1, a) would have a T(S)with the parameters of G. Lemma 2. Zf G is a pseudo-geometric strongly regular graph corresponding to the

parameters s, t, a,then for a clique L in G we have

and if equality holds, then every vertex not in L is adjacent to a vertices of L. Proof. This follows from Lemma 1 applied to the complement of G. 0 The following lemma is known, but not well enough !

Lemma 3. Let G be a pseudo-geometric strongly regular graph corresponding to the parameters s, t. a. Let 2 be a collection of cliques of G,each of size s + 1, such that (i) 1 2 1= ( t + l)(st + a ) / a , (ii) each edge of G is in at least one element of 2. Then the incidence structure with the vertices of G as points, the cliques of .Y as lines and inclusion as incidence is a pg(s + 1, t + 1, a). Proof. Since (";')I2 equals 1 t h e number of edges of G, each edge is in exactly one element of 2. Since G is pseudo-geometric this implies that axioms (i) and (ii) of a partial geometry are satisfied. The third axiom follows from Lemma2. 0

W.Haemers, J.H. DM Lint

208

In our construction of a partial geometry pg(9,8,4) a function the 4-subsets of points of PG(1,8) plays a central role.

Q

defined on

Deflnition. Let PG(l,8) be described in the usual way as FEU {m}. Define

~ ( ab,,c. d ) :=

(a b c +aa+bbd ++ acc+d d+ b c d

First observe that Q depends only on the 4-tuple {a,b, c, d } and not on the order of the elements. Furthermore q ( l + a, 1+ b, I + c, 1 + d ) = 1 + ~ ( ab,,c, d ) , ~ ( a - 'b-', , c-', & I ) = (cp(a,b, C, d ) ) - ' , cp(aa, ab. ac, a d ) = acp(n, b, c, d ) if a # 0 .

Since pSL(2,8) is generated by the transformations x -+ x + 1, x + l / x , x + a x we see that cp and PSL(2,8) commute in the above sense. Let V be the set of nine points of PG(1,8). We can consider Q as a function on (y), i.e. the set of Csubsets of V.

Lemma 4. For every x E V, the points of V \ { x } and the blocks { X : Q(X)= x} form the 3-(8,4, 1) design of points and planes of AG(3,2). Proef. W.1.o.g. we can take x = m. The definition of Q implies that the sets X = {a, b, c, d } with Q(X)= m satisfy a + b + c + d = 0, so they are the planes of AG(3,2) in its representation as FE. 0

Lemma 6. If A E (7) and then cp; = I. ploof.

(PA:

V\A+ V\A is defined by Q A ( x ) : = ~ ~ (UA{ x } ) ,

Since pSL(2,8) is 3-transitive on V we may take A = (0, 1 , m}. Then

QA(X) = x4, so q > ( x ) = x.

C d a r y 7. If A E (y), then A and the two orbits of (pA are a partition of V into three 3-subsets.

209

A partial geomem pg(9, 8, 4 )

Lemma 8. If {A, B, C } is the partition determined by same partition.

(pA,

then (pB determines the

Proof. Let x E C, i.e., C = { x , (PA@), (pi@)}. By (ii) of Corollary 5 and by Lemma 6 we have (pB(x)= cpi(x) E C, i.e., C is an orbit of (pB. 0 3. Construction of pg(9,8,4) We first construct a strongly regular graph G with parameters (135, 64, 28, 32). As vertices of G we consider all binary words of length 9 with weight 4 or 8. There are 6)+ = 135 such words. Two words are joined b; an edge iff their distance is 2 or 6. It is elementary to check that this yields the required graph. In fact this description of the graph differs only slightly from the usual description (using the quadratic form xixi). The difficulty in finding the corresponding partial geometry lies in the fact that G has far too many cliques of size 9. These are of four types: Type 1. The rows of (J - Ib, i.e., the nine words of weight 8. Type 2. The rows of a matrix which has the form

c)

There are 9 * (3/2 = 315 of these. Type 3. The rows of a matrix which has the form

(ins)

where all submatnces are 3 by 3. There are @ Type 4 . The rows of a matrix of the form

There are (:) = 84 of these.

- e) = 560 of these.

W. Haumers, J.H. oan Lint

710

The partial geometry which we wish t o construct has 120 lines, so we have to pick 120 out of 960 maximal cliques. We shall take the clique of Type 1 , 63 cliques of Type 2. and 56 cliques of Type 3. It may be interesting for the reader to know that the numbers of types of cliques led us to this choice and subsequently the problem of finding a 'natural' way to pick the required subsets led us t o the lemmas of Section 2. W e identify the nine positions with the set V. There are 9 choices for the first row of a matrix of t y p e 2, depending on the position of the 0. If 0 is in position x we shall take only those partitionings of the remainder into two 4-subsets which correspond to the 3-design of Lemma 4. This yields 63 cliques of Type 2. A clique of Type 3 is chosen only if the three 3-subsets of V (used in forming the matrix) are as in Corollary 7. This yields 2@/3 = 56 such cliques. The cliques chosen above are defined to be the lines. W e now claim that we have constructed pg(9, 8,4). By Lemma 3 it suffices t o show that every edge of G is in at least one of the lines. There are four cases to distinguish. Case I . Two words of weight 8 are in the line of Type 1. Cuse 2. A word of weight 8 and a word of weight 4 with distance 6 have three i's in common. By Lemma 4 these three positions uniquely determine a line of Type 2 containing this edge of G. Cuse 3. Two words of weight 4 with distance 2 look like

1 I 1 1 1 I

I 0 0 1

D

a b

0 0 0 0 0 0 0 0

-.

X

If cp(X)E D the edge is in a line of Type 2 with the 0 of row 1 in position q(X) and X as one of the 4tuples. If p(X) = a, then q D ( b )= a, i.e.. a and b are in the same class of the partition determined by D (Corollary 7). Hence the edge in question is in a line of Type 3. Case 3. Two words of weight 4 with distance 6 look like

-

0 0 1 0 0 1

I 1 1 0 0 0

0 0 0 1 1 I

a h c

D

E

- .

(i) Clearly there is a Type 2 line containing this edgc if pb'(c) E { a , h } ; (ii) Next, suppose that q b ' ( c ) = g for some g E E. Then (iii) of Corollary 5 implies that q ; ' ( c ) E D. Now if q & ( g ) = cpD(c) is in E we are finished because then we have a Type 3 line containing the edge in question. The same is true if

q E ( c )E D. So it remains to check whether it is possible that pD(c)E {a, b } and pE(c)E {a,b}. Assume that pD(c)= a. Then by (iii) of Corollary 5 we have pE(c)= b. Let E = {g, h, i}. The partition determined by qDis {D,{ g , c, a } , {b, h, i}}. By Lemma 8 it follows that p(a,h, &) E {a, c}. On the other hand q ( b . h, &) =

pE(b)= cp&(c) = qE1(c)E D, a contradiction. So the remaining possibility does not occur and the proof is complete.

4. Remarks

The authors of [3] conjecture, and we agree, that the pg(9,8,4) constructed by Cohen, the one constructed in this paper and the one associated with the hyperbolic quadric in PG(7,2) are isomorphic. In fact they consider it likely that pg(9,8,4) is unique. These questions are still open. A nice feature of our description is that one immediately sees that PSL(2,8) is an automorphism group of the geometry which stabilizes a line. If the conjecture above is true the full group should be A,,. Let us consider the dual of pg(9,8,4) (i.e., we interchange the roles of points and lines). This is a pg(8, 9,4) which has the nice property that it admits a parallelism, that is the lines can be partitioned into parallel classes such that Euclid’s axiom holds. Indeed, for each x E PG(1,8) = V define

cx:={x€(Y): p(X)= x } u ( v \ { x } } . Then it is straightforward to verify that the sets C, produce the required partitioning of the points of pg(9,8,4), i.e., the lines of pg(8,9,4), into parallel classes. A rather small candidate for a new partial geometry is pg(5,8,2) with 75 points and 120 lines. Since the geometry has 8 lines per point and the same number of lines as pg(9,8,4) one might hope that such a pg(5,8,2) is a partial subgeometry of the larger one which means that it is obtainable by deleting a suitably chosen set of 60 points of pg(9,8,4) (four on each line). The points corresponding t o four parallel classes in the dual geometry actually have the property that there are four on each line. This construction does not work ! In fact the whole idea has no chance because of the following theorem. Theorem 9. Let S be a pg(9,8,4) and let S’ be a pg(5,8,2). Then S’ cannot be a partial subgeometry of S.

Proof. Let A and A’ be the adjacency matrices of the graphs T ( S ) and T(S‘). Then. A has an eigenvalue 4 of multiplicity 84 and hence rank(A - 41) = 51. If S’

212

W.Haemers, J.H. van Lint

is a subgeometry of S, then A' is a principal submatrix of A and this implies that rank(A' - 41) 6 51. From this it follows that A' has an eigenvalue 4 of multiplicity at least 24 but A' does not have an eigenvalue 4 at all ! Therefore S' is not a subgeometry of

s. n

Department of Mathematics Eindhoven University of Technology Eindhoven. The Netherlands

References [ 11 R.C. Bose. Strongly regular graphs, partial geometries and partially balanced designs. Pacific J. Math. 13 (1963) 38-19, [2] A.M. Cohen, A new partial geometry with parameters (s, I, a)= (7, 8. 4). Report ZN 92/79, Math. Centre. Amsterdam (1979). [3] F. De Clerck. R.H. Dye and J.A. Thas, An infinite class of partial geometries associated with the hyperbolic quadric in PG(4n - 1,2), Europ. J. Combin. 1 (1980) 323-326. 141 W. Haemers. Eigenvalue techniques in design and graph theory, Thesis, Tract 121. Math. Centre, Amslerdam. 1980. [S] W. Haemers. A new partial geometry constructed from the Hoffman-Singleton graph, Proc. Conf. Finite Geometries and Designs, London Math. Soc. Lect. Notes Ser. 49 (1980) 119-127. [h] J.H. van Lint and A. Schrijver. Construction of strongly regular graphs, Two weight codes and partial geometries by finite fields, Cornbinatorica 1 (1980). [7] J.A. Thas, Combinatorics of partial geometries and generalized quadrangles, in: M. Aigner, 4.. Higher Combinatorics (Reidel, Dordrecht, 1977). [ 8 ] J.A.Thas.Some results on quadrics and a new class of partial geometries, Simon Stevin 55 (1981) 129- 139.

Annals of Discrete Mathematics 15 (1982) 213-227 @ North-Holland Publishing Company

HOMOMORPHISM INTERPOLATION A N D APPROXIMATION Z. HEDRLiN, P. HELL and C.S. KO Dedicated to Rofessor N.S. Mendelsohn on the cccasion of his 65th birthday Starting with the Interpolation Theorem for complete graph homomorphisms, we study general homomorphism interpolation. In particular, we give a set of graphs one of which is a homomorphic image of any graph with homomorphisms onto both K, and K.+l. Similar results are given for graphs with homomorphisms onto both K2 and Ch+l. We also study the notion of defect of a mapping between two graphs as a measure of nearness to being a homomorphism. We give a bound for the best defect of a mapping between two graphs, and show that deciding if there is a mapping between two graphsof defect at most k is an NP-complete problem for each k. Compatibility functions, which reflect the fine structure of defects, are introduced and briefly studied.

1. Homomorphisms

Let G and H be graphs. A homomorphism f : G - + H is a mapping V ( G )+ V ( H ) which preserves edges, i.e., such that gg’€ E ( G ) implies f(g)f(g’)€ E ( H ) . Several other definitions of the term have been proposed in the past (e.g., [5,6]), but this definition is generally accepted today as the appropriate kind of mapping between graphs. It was pioneered by Sabidussi [28,29], Hedrlin and Pultr [12, 13, 141, and Hedetniemi [ll].Homomorphisms have been studied extensively, e.g., [4, 8, 10, 12, 16, 17, 18, 19, 23, 24, 30, 311, and have been applied in such diverse areas as Ramsey theory, Ulam’s conjecture, the study of communication networks, data structures, and scheduling [3, 15, 25, 26, 27, 311. In this note we make several observations on two largely unrelated aspects of homomorphisms. In the next section we examine the amount of information on homomorphic images of a graph one can extract from the knowledge of two concrete homomorphic images. This, in a sense, extends and strengthens the Interpolation Theorem of [lo]. As a consequence we obtain some insight into the manner in which homomorphisms can increase the chromatic number. In the last section we study mappings V ( G ) + V ( H ) which are near t o being a homomorphism G + H. Such mappings are of interest if there is no true homomorphism G + H, or if it is too difficult to find one. A measure of ‘nearness’, called the defect, is investigated, and mappings with a small defect sought. We also analyze some additional parameters related to the notion of defect. 213

Z . Hedrlin. P. Hell, C.S. KO

214

Let f : G H be a homomorphism. The graph f(G) with the vertex f ( V(G)) and the edges f ( g ) f ( g ’ ) for g g ’ E E ( G ) is a subgraph of H, called the homontorphic image of G under f. If f ( G )= H, f is called an epirnorphism (a full epimorphism. [ 161). A monomorphism is a one-to-one homomorphism. The notation f : G H indicates that f is a monomorphism, and f : G + H that f is an epimorphism. The abbreviations G - , H, G - H , and G - H usually indicate t h e existence of a homomorphism, monomorphism, and epimorphism of G to H, respectively. N o t e that a homomorphism f : G -+ K , is precisely an n-coloring of G. An epimorphism f : G ++ K , is called a complete coloring [9]. (Thus an n-coloring is complete i f any two color classes are adjacent.) Evidently, if X G = t i , then every n-coloring of G is complete, and G K.. If there exists a homornorphism f : G + H, then XG S x H , because any n-coloring of H, c : H - + K,. can be composed with f to yield an n-coloring of G, c o f : G + K.. Thus G-+ H implies XG s x H . Just as colorings are sometimes described in terms of the partition into color classes they induce, so it is often useful to describe homomorphisms by their associated partitions. If f : G + H is a homomorphism, -I denotes the equiva1enc.e relation o n V ( G ) whose classes are f-’(h), h E V ( H ) . An equivalence relation o n V ( G ) whose classes are independent sets of G is called a congruence on G. Evidently, - f is a congruence o n G, because our graphs have no loops. For every congruence on G, one can define the quotient graph G/- whose vertices are the classes of and in which two classes are adjacent if there is at least one edge of G joining them. There is a canonical epimorphism f : G + G / - taking each vertex o f G to the class of containing it; clearly, is -f. Observe that a homomorphism f : G + H is a monomorphism if and only if --I is the identity relation on V(G), i.e.. every class of is a singleton. An epimorphism E : G + H for which the classes of are all singletons except for one class consisting of precisely two vertices, is called an elementary homomorphism. The existence of an elementary homomorphism G onto H is abbreviated by G A H. Intuitively, an elementary homomorphism is an identification of two (non-adjacent) vertices. Obviously. an elementary homomorphism cannot increase the chromatic number by more than one, i.e.. G A H implies xG <,yH s x G + 1 [lo]. In closing, we remark that an epimorphism between two finite graphs is always a composition of elementary homomorphisms. -+

H

-

-

-

-

-f

--F

2. Interpolation

In this section we shall often assume that G has finite homomorphic images

H I and H.. and will make deductions about further homomorphic images of G.

Homomorphism interpolation and approximation

215

First, we observe that in such circumstances G can be assumed to be finite. Indeed, suppose that HI,H,, H3 are finite and that for any finite G’, G’-W HIand G’-wHz imply G ’ + H 3 . Let G be an infinite graph, and f l : G + H l , f2 : G -W H2 be epimorphisms. The congruence on G with classes C n D, for C a class of -f, and D a class of -h, yields an epimorphism f :G G’, in which G’ = GI- is a finite graph. Moreover, G’+ HI and G‘-wHz, because each class of -f,, or -f2 is a union of classes of -. Thus G‘ + H3, and G --H H3 by composing G-w G’ and G ’ + H 3 . To illustrate the method, we prove the following well-known fact (at least, well-known for finite graphs).

-

Theorem 1 [lo]. If G + K,, and G

-W

-.,.

K,+k for some k a 2, then G -W K,,+l.

Proof. We have just observed that it is enough t o prove the theorem for finite G. Hence any epimorphism f : G + K,+k is a composition of elementary .* * 0 each of which either preserves the homomorphisms, f = E, 0 chromatic number or increases it by 1. Let GI = G, and Gi = E ~ - ~ ( G ~ - ~ ) . Since X G , +=~x ( f ( G ) ) =n + k , for some j , j = 2,. . . ,r, xGj = n + 1, and G ++ Gj + K,,+l. 0

There is an obvious aspect of interpolation to the above theorem. In particular, when k = 2, it assures a homomorphic image (Kn+l)between K,, and K,,+*.We would like to continue the interpolation, and find a homomorphic image between K,, and K,,+l-a ‘K,,+ln’. To make this precise, let n 3 2, and define t o be the class of graphs obtained from K,, by adjoining two new non-adjacent vertices a, 6, and joining a, respectively b, t o some, but not all, vertices of K,, in such a way that every vertex of K,, is adjacent to at least one of a, b (cf. Fig. 1).

Fig. 1. Some graphs in Xd+112.

Note that for any K E X,,+1,2, K + K, (identify a, respectively b, with some vertex of K,, to which it is not adjacent) and KfttK,,+I(identify a with b). Let n a 2.

Theorem 2. If G -W K,, and G -W K,,+l, then G --H K for some K E Xn+lr-. Proof. We may assume that G is finite, and f : G + is a composition of elementary homomorphisms, f = E , o E , - I ~ . . . O E ~ . Let GI = G and G, = E ~ - , ( G ~For - J .some j , xGj = n and X G , +=~n + 1. Let - c be the congruence on

Z . Hedrlin, P. Hell, C.S. KO

216

G, corresponding to some n-coloring c : G,-,K,,. We shall describe a congruence on G, with GJ- E Since G -++ G, (by . E , ) and G, --w GI/--(by the canonical epimorphism), this will prove the theorem. G,+Iis obtained from GI by identifying two vertices, say g, and g?. Since XG,,, = n + 1, any n-coloring of G,, and in particular c, must assign different colors to gI and gz. or else their common color could be kept in GI+,.Let the classes of - c be C,, C,. , . . , C,, and g, E C,,gZEC2. The congruence is defined to have the classes {g,}, {gd, C, - {gl}, C2- {g2}, C,, . . . , C,,. Any two of the classes C, - {gl}, C2- {g2}, C,, . . . , C,, are adjacent, or else G, - {gl, g2} would be (n - 1)-colorable contrary to the fact that G,,, is not n-colorable. For the same reason C, - {gl}# 8. C2 - {g2} Z 8. Thus CI- {gl), C2 - {g2Ir C,, . . . , C,, form a copy of K , in G,/-. The vertex g, is not adjacent to CI- {g,}, but is adjacent to C2- {g2} or else moving g, to C2would change c to an n-coloring of GI assigning the same color to g, and g,. Similarly, the vertex g2 is not adjacent t o C, - {gJ but is adjacent to C, - {g,}. Finally, for any i 2 3, C, is adjacent to gl or g2, or else moving both gl and g2 to C, would again change c to an n-coloring of G, in which gl and g2 have the same color. Therefore G,/- E 3y,+,i2.

-

-

Theorem 2 with n = 2 can be deduced from 1161. Theorem 2 can be viewed as a statement o n factoring homomorphisms. An n-chromatic graph G which admits an epimorphism G -,., Kn+,admits an epimorphism G K,,,, which can be factored into a X-preserving epimorphism and an elementary homomorphism. The following corollary states this more generally.

-

Corollary 3. Zf xG s n and G xH = n.

--c)

K,,,,, then G - H f , Kn+,for some H with

If XG = m 6 n, then G K , and, by Theorem 1, G --H K,,. By Theorem 2. G -,., H for some H E X,+ln; thus xH = n and H 2 K,+I.

prod.

--.I

Theorem 4. If G is a bipartite graph and G --H C Z k + l , then G --w H f, c 2 k + 1 for some bipartite graph H. We shall prove that G n K 2 and G --H c 2 k + 1 imply G + P2k+z. where Pzk+? is t h e path with 2k + 2 vertices (hence of length 2k + 1). Then clearly P2k+2$ CZir+, by identifying the endpoints. We may again assume that G is finite, and f : G --n c 2 k + 1 is a composition of elementary homomorphisms, f = F , 0 E , - , o . * .o Let GI= G and Gi = E * - , ( G ~ -Let ~ ) . j be the largest subscript such that Gj is bipartite, and let g,, gz be the vertices of Gj identified by E,. Since GI+,contains an odd cycle, g, and g2 are connected in Gi by a path P of odd length. We observe that the distance between gl and gz in Gj is at proof.

Homomorphism interpolation and approximation

217

least 2k + 1. Otherwise a shortest path between gl and gzin Gi, which must be of odd length because P is of odd length and Gj is bipartite, would be mapped by E~ to an odd cycle in Gj+,whose length is smaller than 2k + 1. Since a homomorphic image of an odd cycle of length < 2 k + 1 must contain such a cycle, we would obtain such a cycle in CZk+1,a contradiction. Let C be the component of Gj containing g , and g,. We have just proved that the diameter of C is d 2 2k + 1. Let go be a vertex of maximum eccentricity of C (cf. [9, p. 35]), Do= {go}, and Di, 1 S i S d, be the set of vertices of C of distance i from go. It is easy to see that each Di is an independent set in C and that Di is adjacent to Di-, and Di+,but no other Di. (Doadjacent only to D1 and Dd adjacent only to Dd-1.) Thus Do,.. . , D d are the classes of a congruence on C whose quotient CI- is P d + l . Since C+Pd+I and P,-"Pl-l (by an evident elementary homomorphism), C + P2k+*.It is easy to extend such an epimorphism t o Gi--H PZk+2 by taking a two-coloring of Gi - C and mapping all red vertices to the first vertex of Pzk+2 and all blue vertices to the second one. Finally G --H PZk+* because G --n Gj by ej-, E ~ . Note that G G', X G = xG' - 1 does not in general imply the existence of an H with X H = XG and G H InG', even when G' is critical (i.e., x(G' - g) < xG' for all g E V(G')).This can be seen by taking G' = W,, the 5-wheel, and G the graph obtained from the 5-cycle C, by adjoining 3 new vertices a, b, c and the 10 edges a l , a2, a4, b l , b3, b5, c2, c3, c4, c5.

-

0

-

0

--f*

++

Consider a finite graph G with XG = n. If f : G + Kn+l,then f = E, 0 * . . o E , and exactly one E~ increases the chromatic number. Theorem 2 asserts that the increase can be postponed till the last elementary homomorphism, i.e., that for some f' : G --n Kn+l,f' = E ~ O .. . o E ; , it is E : which increases the chromatic number. (Similar interpretation applies to Theorem 4.) If f : G ++ Kn+,, then f = E, and exactly two E ' S increase the chromatic number. However, it is no longer always possible to arrange things so that they are the last two elementary homomorphisms: In Fig. 2, it is easy t o verify that any epimor) ) 3. On the other hand, if in some f the phism E ; O E ; O E ; : G --H K4 has x ( E ; ( G = two E ' S increasing the chromatic number are consecutive, then in some other f' = E ; O 0 E ; : G ++ Kn+2 the increase occurs in E : , E ; - , . The following theorem states this more generally.

-

0

-

0

AG

€3

I

.

,

-_- , c

€2

Fig. 2.

G-K,.

E , ~ E ~ ~ E , :

2. Hedrlfn, P. Hell, C.S. KO

218

+.

Theorem 5. Let XG = n and assume that f : G K,,, is the composition f = fu 0 F, 0 F , - , 0 . . . o fa where each E,, i = 1, . . . , r, is a n elementary homomorphism increasing the chromatic number. Then there is a n f‘ : G-w K,,, f ’ = C:OE;. . . O E ’ , o f m where each E : , i = 1, . . . , r, is a n elementary homomorphism iwreectsing the chromatic number. 0

0 .

Proof. Note that fu, fw, fb preserve the chromatic number. Also note that Corollary 3 implies Theorem 5 for r = 1. We may assume that f = fw E, 0 . . * o el, or else we could replace G by f,(G). Let GI = G and G, = E,.,(G,-,); let the vertices of GI identified by E , be g, and g:. Since e = F , O . . ’ O E , increases the chromatic number of G by r, there are at least r vertices u in e ( G ) = G,,, with le-’(L>)l> 1. This implies that E ] , . . . , E , identify disjoint pairs of vertices of G, i.e., that gl, . . . . g,, g ; . . . . , g : are distinct. Let S = {gl,. . . ,g,, g ; , . . . , g : } . Let CI,Cz,. . . C,, be the classes of some n-coloring G 4 K,,. Consider the congruence on G whose classes are {gl}, . . . ,{g,}, {gi}, . . . ,{g;}, CIS, . . . , C,, - S. Let G’ = G/- and f : : G --n G’ the canonical epimorphism. Clearly, xG’ = n because each g,, respectively g : , is not adjacent to some C, - S. (Namely for j such that C, contains g,, respectively g : . )Let E : identify g, and g: and let f = E : O . . . o E ; o f : ; we shall show that f’(G)= K,,,. Since xG,, = n + r. any (n + r)-coloring of e( G ) = G,,I is complete. Let g’i be the vertex of e ( G ) resulting from t h e identification of g, and g: by E l . Then {g:}. . . . { g ; } , CI- S, C2- S, . . . , C,,- S are the classes of a complete (n + r)coloring of e(G), and consequently { g , , gi}. . . . , (g, g:}, Cl - S. C, - S, . . . , C,, S are the classes of a complete (n + r)-coloring of G. On the other hand, they are precisely the classes of -,.. Thusf’(G) = K,,,. Consequently, each E : increases the chromatic number by 1. 0

-

.

.

-

A strongly complete n-coloring of G is a coloring c : G K, in which every color class contains a vertex adjacent t o all other color classes. If XG = n, it is easy to see that every n-coloring of G is strongly complete. Conversely, if G admits complete n-colorings and each complete n-coloring is strongly complete. then XG = n by Theorem 2. (Every K E X(n-l)+l,2 has a complete n-coloring which is not strongly complete; if G K. then G also admits such a coloring.) This observation is generalized to multicolorings in [21]. --H

3. Approximation

In this section we shall consider only finite graphs. In practical applications, a homomorphism f : G + H may correspond t o a desired arrangement of objects, or schedule of activities. The general problem of existence of a homomorphism f : G + H is NP-complete, because it includes the question of

Homomorphism inferpolation and approximation

219

-

n-colorability of G (taking H = K,,) [20]. Of course, in special cases the existence of a homomorphism f : G + H is easy to determine. For example, among bipartite graphs always G H, because G + K 2 - H. For trees, there are polynomial time algorithms to determine whether T ++ T’ [22]. Still, even for trees, it is an NP-complete problem to determine whether T + T‘ [7]. If there does not exist a homomorphism f : G -+ H, or if it requires too much effort to find one, we may wish to find a mapping f : V(G)-+ V ( H ) which is near to being a homomorphism. One way to measure such nearness is the number of edges to be removed from G and/or added to H to make f into a homomorphism, f : G --* H. To be more specific, let G and H be graphs, and let

G’ = {G’1 V(G’) = V ( G ) ,E(G’) E ( G ) , (E(G)- E(G’)I S i } ,

H, = {H’ I V ( H ’ )= V ( H ) ,E ( H ’ )2 E ( H ) , IE(H’)- E(H)I s j ) . The set Homj(G, H) consists of all homomorphisms f : G‘+ H’ for all G‘ E G’ and H E 4.Thus Homt(G, H) is the set of all homomorphisms of G to H, and f E Homj(G, H ) if and only if one can remove some S i edges of G and add some S j edges to H in such a way that f becomes a homomorphism. Let f : V ( G ) + V ( H ) be any mapping. We define the negative defect, d - ( f ) = min{i 1 f E Homb(G, H ) } , the positive defect, d+cf)= minG I f E Hom:(G, H)}, and the defect, dcf) = min{i + j I f E Homf(G, H ) } . Note that when discussing the various notions of defect for a mapping f , it is important to know not only V(G) and V ( H ) ,but also G and H ; for instance any f : V ( G ) + V ( H )is of defect 0 as a mapping from the graph on V ( G )which has no edges. Therefore we shall always discuss the defects of a mapping between two graphs. In order not to confuse such mappings with homomorphisms, we shall use the notation f : G -+ H to indicate that f is a mapping of the vertex-set of the graph G to the vertex-set of the graph H. The notion of negative defect was used in an inclusion-exclusion argument by Muller to verify the edge-reconstruction conjecture for a large class of graphs [25]. From now on, we shall always assume that G and H have the same number of vertices, denoted by p.

Theorem 6. Let xG = n, XH = m, and p = 9m + r for non-negative integers 9 and r, 0 =Z r < m. (a) The minimum positive defect of any f : G-+ Kpis (1). (b) If K,,, is a subgraph of H, then the minimum negative defect of any f :K,-+Hism(j)+qr. Proof. (a) The minimum positive defect of any f : G -+ Kp is the minimum j such that G + G‘ for some G‘ with p vertices and j edges. Such a G‘ is then a

2. Hedrlin. P. Hell, C.S. KO

220

‘minimum range’ of G. Since xG = n, G - , K,,, and K,, has (1) edges. Moreover, if G 4 G‘, then x G ’ n~and hence V ( G ’ )can be partitioned into at least n classes with at least one edge between any two classes. Thus G‘ has at least (?) edges. (In fact, it is not possible that G’ has precisely (1) edges unless G‘ = K , ; hence Kn is the only minimum range of G. The analysis of minimurn ranges is more interesting for digraphs [18].) (b) The minimum negative defect of any f : K p+ H is the minimum i such that H ’ - H for some H’ with p vertices and (5)- i edges, i.e., (5) minus the maximum j such that H ‘ + H for some H’ with p vertices and j edges. Such an H‘ is then a ‘maximum domain’ of H. Since K,,, is a subgraph of H, H’+ H if s rn. Thus the maximum domain H’of H will be the largest and only if xH’ m-partite graph with p vertices. Hence H’ is the complete m-partite graph with all parts as equal as possible. If p = 9m + r, 0 G r < m, then r parts will have 9 + 1 vertices and m - r parts 9 vertices. Therefore, the minimum negative defect is the number of edges in H ’ , which is r(q;‘)+ ( m - r)(i) = m(;)+ 91. Since adding edges t o H and removing edges from G cannot increase any kind of defect, we have the following corollary.

Corollary 7 . Under the circumstances of Theorem 6, there always exists a mapping f : G--*H of positive defect at most (1) and a mapping f’: G - +H of negative defect at most m(f) + 91. Both bounds are bestpossible, i.e.. are achieved for some pairs G, H. Let No= (0. I , ? , . . .} and G, H be fixed graphs with p vertices. We shall define a function CG.”: No+No which captures the tradeoff, in finding a homomorphism of G to H, between the deletion of edges from G and the addition of edges to H: For any j E No,

C , , ( j )= min{i 1 Homj(G, H) # 8) Thus C , & ) is the smallest number of edges whose deletion from G allows a homomorphism to H with some j edges added, i.e., C G , , ( ~ ) = min{d-Cf) I f : G -+ H’ with H‘ E H,}. Also note that min{d(f) I f : G - +H} = minG + C,,,(j) I j E No}. Clearly, each C,,is a decreasing function No-, No.

Lemma 8. Let 1

In particular, if p

m

=

dp

qm,

and p

= qm

+ r, 0 < r < m. If

(7) s j < (“‘;I),

then

Homomorphism interpolation and approximation

22 1

Proof. Let (7) s j < ('"Tl).As observed above, we seek the minimum negative defect d - ( f ) of any f : Kp-+H with IE(H)(S j . Since such an H has fewer than ('"$I) edges, it is m -partite. Hence, any G with G + H must also be m -partite and therefore has at most m($)+qr edges. Thus d - ( f ) 3 m ( f ) +qr for any f : Kp-+ H. Let H be the disjoint union of K , and Kp-,,,.According to Theorem 6(b) there is a mapping f : Kp-+ H of negative defect m (5) + qr. This proves the lemma. (We note that one can prove that, for in 2 5, a graph H with fewer than ('"TI) edges which does not contain K , is (m - 1)-colorable. Indeed, if X H = m, then any m-critical subgraph of H would have at least m + 2 vertices [2, p. 1211 and at least i((m - l)(m + 2) + ( m- 3))3 ('"l') edges [ 1, p. 2361. Thus the smallest negative defect can only be attained by placing the j edges so as to form the largest possible complete subgraph in H.)

Theorem 9. ntere always exists a mapping f : G-+ H of defect at most

and the bound is asymptotically best possible, because the minimum defect of a mapping f : Kp-+ k7, is greater than 3(32)-'/3p4"- ;p. Proof. We computed CKP%in Lemma 8. From there we see that I j E NO}occurs when j = (7) for some min{dCf) I f : Kp-+Ep}= minG + CKPzp(j) m = 1, . . . ,p. Thus we need to minimize the function

where p = mq + r, 0 s r < m, and 1s m s p . Since q = I p / m ] , (p - m ) / m < q , s p/m, and

Therefore,

77

& I . .l

Z. Hedrlin. P.Hell, C.S. KO

The minimum of Am2+{p’/m occurs at m = 2-1/3p2/3, and is 3(32)-’/3p4/9.Hence the minimum of f ( m ) is strictly between 3(32)-”3p4’3-zp and 3(32)-”3p4/3f $p. It is clear that the minimum defect of any f’ : G -+ H is at most the minimum defect of a mapping f : K,, -+ K,. We have found that on the one hand, it is difficult to determine if there is a mapping f : Ci--.H of defect 0 (i.e.. a homomorphism f : G + H), and on the other hand, there always is a mapping f : G - + H of defect at most 3(32)-”3p”3+;p. It is conceivable that for some large fixed k, the problem of existence of a mapping f : G -+ H of defect k becomes easy. That, however, is not the case. For k E No, we denote by DEF(k) the decision problem: Given G and H (with the same number of vertices), does there exist a mapping f : ( 3 - + H with d ( f ) c k ?

Theorem 10. DEF(k) is NP-complete for any k E No. Before we embark on a proof. we introduce some concepts that shall also be used later. Let S be a graph, t 3 1 a n integer. A sequence of vertices xo, xl,. . . , x, of S is a power-t-path from xo to x, in S. if z 2 t, and ax, E E ( S ) whenever 0 < Ii -jlc t. A graph S is power-t-connected [17] if any two vertices of S are joined by some power-t-path in S. We observe that if S is power-t-connected, r 2 2. then for any edge e of S, S - e is power-([ - 1)-connected: Consider two arbitrary vertices x and y of S and a power-t-path x = xg, x,. . . . , x, = y joining them. The same sequence of vertices is a power-(t- 1)-path in S - e, unless e = x,xb with a < h and a f 0 or 6 f 2. Since the situation is symmetric, we assume that a # 0. A power-(1 - 1)-path joining x and y in S - e can be obtained by removing x, from x,), xi, . . . , x,. We also note that the homomorphic image of a power-r-connected graph is likewise power-t-connected. Finally, observe that a power-t-connected graph contains K,,,. Let S, be the graph with vertices 0, 1, . . . , 3 t + 3 and edges ij, for li - jl S f, and 0. 3t + 3. It is easy to see that ,yS, = t + 2 and, in fact, S, is ( t + 2)-critical. Also. S, is power-t-connected, and hence S, - {el, . . . , e,} is power-(t - s)connected, for any set of .F edges { el , . . . ,e,} of S,.

Proof of Theorem 10. Evidently, DEF(k) E NP. Since DEF(0) is the problem of existence of a homomorphism G + H, it is NP-complete. (The fact that G and H have the same number of vertices poses n o problem, since G-+ K, U&, if and only if G-, K,. which we have earlier observed to be NP-complete.) We shall exhibit a polynomial reduction of DEF(k) to DEF(k + 1); then each DEF(k) is NP-complete by induction. Given graphs G and H with the same number o f vertices, we shall construct, in polynomial time, graphs G’ and H’

223

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with the same number of vertices, such that there is a mapping f : G - +H with defect d ( f )=sk if and only if there is a mapping f’ : G ’ - + H ’ with defect d ( f ‘ ) s k + 1. Let I V(G)( = I V(H)l=p and t = p + 2k + 3. Let G’ be obtained from G by attaching a copy of S,, by vertex 0, at every vertex of G, and adding to one copy of S, the additional edge t + 2, 3t + 3. Let H’ be obtained from H by attaching a copy of S , by vertex 0, at every vertex of H (cf. Fig. 3). Any mapping f : G - + H of defect dCf)s k can be extended to a mapping f’ : G I - + H ’ of defect =sk + 1, by mapping the vertex i of the copy of S, attached at g E V ( G )to the vertex i of the copy of S, attached at f(g) E V(H). Conversely, consider a mapping f’ : G’-.+H’ of defect d(f‘) S k + 1. Let f’ E Homj(G’, H’) with i + j = d(f‘). Thus f’ is a homomorphism f ’ : G”+ H” where G” is obtained from G’ by deleting some i edges and H” from H’ by adding some j edges. Let V be the set of vertices of any one copy of S, in G’. Since S, is power-t-connected, and t - i 3 t - k - 1 = p + k + 2, V induces in G“ a power-@ + k + 2)connected subgraph. Therefore f’(V) also induces a power(p + k + 2)-connected subgraph of H”. Note that H’ is obtained from H” by deleting j S k + 1 edges-thus f’(V) induces in H’ a power-@ + 1)-connected subgraph. It follows that f’(V) is a subset of some copy of S, in H’. Hence we can define a mapping f : G - + H by setting f(g), g E V ( G ) ,equal to h if the copy of S, at g is mapped by f’ into the copy of S, at h. It remains to show that d(f)=sk. Note that every edge of G’ is either an edge of G or an edge of a copy of S,. Of the i edges removed from G’, let il be in the copies of S, and i2 in G ; of the j edges added to H’ let jl be added to the copies of S, and j z between the copies. (Note that this latter category includes edges added to H.) Thus i, i2 = i and jl + j 2 = j. Then il + j l 2 1 because there is no homomorphism of the special copy of S, with the edge t + 2, 3t + 3 t o S,.(There is obviously n o monomorphism, and xS, = t + 2 while X S z s t -t 1for a subgraph S of S, which does not include all vertices of S,.)Thus d ( f )S i2 + j 2 = (i + j ) - (il + j , ) S d ( f ’ )- 1s k .

+

5

5

G’

H‘ Fig. 3. An illustration of G , H’ with t = 2 (in actual constructions I = p + 2k

+ 3).

Z. Hedrlin. P.Hell, C.S. KO

224

Our final observations concern the functions C,, It is obvious from the definition, that each function C G ~ H: No+No is decreasing. Of course, it is not strictly decreasing. in fact there is an s E Nosuch that C G , H ( t ) = 0 for all t > s. The unique smallest s with this property is called the support of C , H . We shall prove that every decreasing function c : No-+Nowith finite support is equal to CG,Hfor some graphs G, H . The first step of the proof consists of proving the assertion for digraphs. Homomorphisms of digraphs are defined in the expected manner, as mappings of vertices which preserve t h e arcs; G' and Hj are defined as for graphs, except that we count the arcs added/deleted, rather than the edges. The definitions of Homj(G, H ) and C , , then apply directly.

Theorem 11. Each decreasing function c : No-+No with finire supporr is equal C& for some digraphs G. H .

To

Proof. Let 7' be the digraph with vertices 1 , 2 , . . . , k, and arcs ij for i < j . (Thus Tk is the transitive tournament of order k.) For i = 0, 1 , . . . , k - 1, let T i

be obtained from Tk by the addition of the vertex 0 and the i arcs . . , Oi. Thus TO, is Tk with the isolated vertex 0. We claim that the function C,-;,T;(j)has value 1 for j s i - 1 and value 0 for j i. Indeed, T i - 01 + TO, (by identifying 0 and l), thus C T ; , T i ( j ) S 1 for all j . Moreover C T ; , T i ( i ) = 0, because T i with the additional edges 01,. . . , O i allows a homomorphism (in fact, isomorphism) from TI. It remains to show that if T is obtained from Tk by the addition of at most i - 1 edges, then T i tr T. This follows easily from the observation that T i does not contain two arcs ab, a f b f with none of the arcs aa', a'a, bb', and b'b. (No two arcscan then be identified by a homomorphism and T i + T implies that T i has at least as many arcs as T.)

01,0?,.

Let c : hi,,-+ Nobe a decreasing function of support s. We may assume that c is not identically zero, or else CT,,J, = c. Thus s 2 1; let k = s + 1. We define

G

7

~ ( -k2)Ti-I U(c(k

- 3 ) - c ( k - 2))T:-*U *

*

U(c(0) - c(1))T:

where U is the disjoint union and xT = T U T U . UT, x times. Since c(0) # 0, not all coefficients in the expression for G can simultaneously be zero. Let H be the disjoint union of T! and (c(0)- l)(k + 1) isolated vertices; evidently, 1 V(G)I = 1 V(H)I. The value of CG,,(j)is the minimum number of arcs that need to be removed from G to allow a homomorphism to some H' obtained from H by the addition of some j arcs. It is easy to deduce from our calculations of C T L that ~,

Homomorphism interpolation and approximation

225

if j s k - 2, and CG,,(~) = 0 = c ( j ) for j 3 k - 1 = s. (A crucial observation for the former conclusion is the fact that the same j edges u l , u 2 , . . . , uj added to H allow a homomorphism to H from each of T i , T:, . . . , T i . ) One may require G and H to be connected (cf. Fig. 4): It suffices to add to G one new vertex w and arcs wk to all vertices k of the copies of T i , and to H one new vertex z adjacent to k and all isolated vertices. It is easy to see that this transformation does not affect C,,.

A+

z

w

1 3+ 4 5 . 3

0

0

1

0

G

Fig. 4. Connected digraphs G, H, yielding the function c defined by c(O)= 3, c ( l ) = 2, c ( j ) = 0 for j 3 2.

Corollary 12. Each decreasing function c : No+ No of finite support is equal to CG,H for some (conneckd) graphs G, H.

Proof. Let G', H be digraphs with C G , , H '= c, and let s be the support of c. We may again assume that s 1. Let t = 2s + c(0) and let G be obtained from G' by replacing each vertex by a copy of Sr and replacing each arc xy by an edge joining the vertex t + 2 of the copy of S, which replaces x, with the vertex 3t + 3 of the copy of S, which replaces y. Let H be obtained from H by the same construction. We claim that C , , = CG',w (= c). (Note that since G', H' have the same number of vertices, so do G and H. Moreover, if G' and H' are connected, then so are G and H.) Clearly, for any mapping f E Homj(G', H'), there is a corresponding mapping f € Homj(G, H), taking the vertex i of the copy of S, which replaces x to the vertex i of the copy of S, which replaces f ( x ) . Thus C , H ( j )s Cc,w(j) and CGvH(j) = 0 (= CG,H,(j))for j 2 s. On the other hand, for j < s, the addition of any j edges to H results in an H in which only subsets of the copies of Sr are

2x7

Z . Hedrlin. P. Hell. C.S. K O

power-2s-connected. (Any set of vertices which is power-2s-connected in I? is j ) < CG..Hs( j ) = c(j ) s power-(s + 1 )-connected in H . ) If. for some j < s. CG,H( c(0) then some G obtained from G by the deletion of fewer than c(0) edges would allow a homomorphism into some fi. Since in any such G, the copies of S, remain power-2s-connected, they would be mapped i n t o copies of S,. It is now easy to mimic the situation in G' and H'-adding some j edges to H' and deleting some C c ; , H ( j < ) C GH . ( j ) e d g efrom ~ G' resulting in a homomorphism, contrary to the definition of C ( ; . . H . .Hence CG,H = CG..H. = C. Charles University Prague. Czechoslovakia Simon Fraser University Burnaby Vancouver, Canada Rutgers University Newark. NJ. USA

References 111 B. Hollobas. Extrcnial Graph Theory (Academic Press. New York. 1978). 121 J.A. Bondy and U.S.R. Murty. Graph Theory with Applications (Elsevier. New York, 1976). (31 S.A. Burr. P. Erdiis and L. Lovbz. On graphs of Ramsey type. A n Combin. 1 (1976) 167-1YO. 141 V. Chviital. P. Hell. L. Kutera and J. NeSetiil. Every finite graph is a full suhgrdph of a rigid graph. J. Combin. Theory I! (1971) 284-286. IS] K. (5ullk. Zur Theorie der Graphen, casopis Pest. Mat. 83 (19%) 13>135. 161 G.A. Dirac. Homomorphism theorems for graphs. Math. Ann. 153 (1%) 69-80. 171 M .Farbcr. personal communication. 1970. [X] M Farzan and D.A. Waller. Kronecker products and local joins of graphs. Canad. J. Math. 29 (lU77) 75.5-269 [S] F. Harary. Graph Theory (Addison-Wesley, Reading, MA. 1972). [lo] F. Harary. S. Hedetniemi and G . Prins. An interpolation theorem for graphical homomorphisms. PortuEal. Math. 26 (1067) . , 45+%2. S.T.Hedetniemi. Homomorphisms of graphs and automata. Tech. Kept. 03105-44-T.University of Michigan. 1966. %. Hedrlin. Extensions of structures and full embeddings of categories, Actes, Congr. Internat. Math. Nice (Gauthier-Villars. Pans, 1971) pp. 31S-322. 2. Hedrlin and A. Pultr. Symmetric relations (undirected graphs) with given semigroups, Monatsh. Math. 60 (1965) 318-322. Z . Hedrlin and A. Pultr. On rigid undirected graphs, Canad. J. Math. 18 (1966) 123771242, G.H. Heil. Structure in social networks. Harvard University Department of Sociology Rept., 1973. P. Hell. Retractions de graphes, Ph.D. thesis. Universitt de Montrtal. 1972. P. Hell. On some strongly rigid families of graphs and the full embeddings they induce. Algebra Universalis 4 (1974) 108-126.

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[It?] P. Hell, An introduction to the category of graphs, Ann. N.Y. Acad. Sciences 328 (1979) 120-136. [I91 P. Hell and D.J. Miller, Graphs with forbidden homomorphic images, Ann. N.Y. Acad. Sciences 319 (1979) 270-280. [20] R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller and J.W. Thatcher, eds., Complexity of Computer Computations (Plenum, New York, 1972) pp. 85-103. [21] C.S. KO, Broadcasting, graph homomorphisms, and chord intersections, Ph.D. thesis, Rutgers University, 1979. [22] D.W. Matula, Subtree isomorphism in O(n5’2),in: B. Alspach, P. Hell and D.J. Miller, eds., Algorithmic Aspects of Combinatorics, Ann. Discrete Math. 2 (1978) pp. 91-106. [23] E. Mendelsohn, Full embeddings and the category of graphs, with applications to topology and algebra, Ph.D. thesis, McGill University, 1968. [24] D.J. Miller, The categorical product of graphs, Canad. J. Math. 20 (1968) 1511-1521. [25] V. Muller, The edge reconstruction hypothesis is true for graphs with more than n lo&n edges, J. Combin. Theory Ser. B 22 (1977) 281-283. [26] J. NeSetiil and V. Radl, The Ramsey property for graphs with given forbidden complete subgraphs, J. Combin. Theory Ser. B 20 (1976) 243-249. [27] J.L. Pfaltz, Graph structures, J. Assoc. Comput. Mach. 19 (1972) 911422. 1281 G. Sabidussi. Graph theory, Tulane University, Mimeographed notes, 1957. [29] G. Sabidussi, Graph derivatives, Math. Z. 76 (1961) 385401. [x)] D.A. Waller, Pullbacks in the category of graphs, in: C. St. J.A. Nash-Williams and J. Sheehan, eds., Proc. 5th British Combin. Conf. (Utilitas Math., Winnipeg, 1976) pp. 637-642. [31] D.A. Waller, Products of graph projections as a model for multistage communication networks, Electron. Lett. 12 (1976) 206-207.

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Annals of Discrete Mathematics 15 (1982) 229-238 @ North-Holland Publishing Company

PROLONGATION IN m -DIMENSIONAL PERMUTATION CUBES Katherine HEINRICH Dedicated to N.S. Mendekohn on his 65th birthday We show that under certain conditions an m-dimensional permutation cube of order n can be prolongated to an m-dimensional permutation cube of order n + 1.

1. Introduction and definitions All definitions are consistent with those of DCnes and Keedwell [4].Let A("') be an m-dimensional n x n x . . x n array, the Celts of which will be denoted ( i l , i 2 , . . . , i m ) where 1 s ij S n for 1 S j S m. We say that A'") is based on the integers 1 , 2 , . . . , n if each cell of A(m)contains one of 1 , 2 , . . . , n and we denote by ail,i2 , . . . ,i,the integer in cell ( i l , i2, . . . , i m ) . Thus we shall write A(")=( a . . , . . _i,). , A column in A'") is a set of n cells Cfl.i, ,.... i,-l,i,+l , . . . , im . . . . . { ( i l , i 2 , . . . , is-l, is, is+l, . . . , 2"): 2 1 , 12, . . . , is-1, zs+l, . . . , i, are fixed and 1 is d n}. Such a column will be said to be of type s. Note that A(")has n m columns. A column is latin if the integers in the cells of the column are a permutation of the integers 1,2,. . . , n. An m-dimensional permutation cube (m-DPC) A(")= (all.Q,. . . ,i,) of order n is an n X n x . . X n m-dimensional array based on the integers 1,2, . . . , n with the property that every column is latin. A transversal T'") in an m-DPC of order n is a set of n cells {(ill, i12,. . . , i l " ) . . . ( i 2 1 . i22, . . . , i 2 m ) , . . . , ( i , , ~ ,1.2, . . . , i,)} such that each ilk for k fixed and 1 =sj s n, is distinct; as are the integers in these cells. If an m-DPC of order n, A'") = (ail,h, . . . , i m ) has an rn-dimensional subarray B'") = (bjIn,,, j m ) which is itself an m-DPC or order 2 then we say that B(")is an intercalate of A'"). A second m-DPC of order n, denoted I(A'")) is obtained from A'") by inverting the intercalate B'");that is, assuming B(") is based on the integers k and l we invert B(")by replacing each k by l and each 1 by k. It is clear that I(A("))is an m-DPC of order n. Given an m-DPC A(") of order n, if we can obtain from it an m-DPC of order n + 1, denoted P(A(")),simply by the addition of a new integer n + 1,we shall say that A'") has been prolonguted. It is well known that if A(")is a 2-DPC (latin square) of order n which has a transversal, then A'") can be prolongated.

-

,,

229

2.w

K. Heinrich

This procedure for latin squares, first used by Rruck (31 and later called prolongation by Belousov [ I ] , will be given in Lemma 1.1. Such prolongation has been further studied by many authors (see [4]). We shall then, in Lemma 1.2. give sufficient conditions under which a 3-DPC can be prolongated and follow this by an example. Section 2 begins with a particular case of prolongation in 4-DPCs. Sufficient conditions are then given to allow prolongation in an rn-DPC. In Section 3 we make some observations regarding the requirements for prolongation in M dimensions.

Lemma 1.1. A 2-DPC which has a transversal can he prolongated.

Proof. Let A ( ’ ) = (a,,,,>)be a 2-DPC of order n with a transversal T(2).Let = (bzl,J be an ( 1 1 + 1 ) x ( n + 1) array with bll.Q = 1 s il, i 2 S n. If (r, s) E T”] put b,, = n t 1 and b,,+l = b.+,., = arS.Finally put h m + l . n=+nl + 1. Clearly is a 2-DPC of order n + 1 and so B(’)= P(A(’)). 0

Lemma 1.2. A 3-DPC A(’)= (a,l,Jsatisfying the following conditions can be prolongated : ( 1 ) Each of the 2-DPCs A::) = ( a,I J, where a , I 12 = a ,1,12,13has a transversal T)f’ for each il, 1 c i 3 s n, and

(2) The cube A‘”’ has a frnnsversal T‘” and if (i,, i2, i3)ET”), then ( i l , i2)E T;;! (3) Ler A(” = ( a 11,,2) where if (il, i ~ E) T::),then a , I , l 2 = a l , . , 2 , 1 3 . We require fhatA(*) be a 2-DPC of order n.

Proof. Let A(’)= ( u , ~ , ~be . , , a) 3-DPC of order n satisfying the above conditions. In A(3’ define Alf) and Ti;), 1 s i3 s n, F3)and a 2-DPC A(2)as above. Let T”)= {(i,, iz): (i,, i,, i?) E T(3)}which is clearly a transversal of A(’). Using the transversals T$t)and T’’)prolongate each of A$:)and A(’) as in Lemma 1.1. Now, in P(A$:)),if (i,, i,, i3) E T(3)invert the intercalate consisting of the cells (il. i?), (i,, n + I), (n + I , iz) and (n + 1, n + 1); so obtaining WW;))). is a 3-dimensional (n + 1) x (n + 1) x ( n + 1) array Suppose that B(3)= (b,,.Q,,3) defined as follows. Denoting by BI:’ the 2-dimensional subarray defined by the cells { ( i t , iz, i3): 1 C il, i2 s n}. for fixed i3, let B$:) = Z(P(A$f))),1 5 i3 =sn, and B(,i1= P(A(*)).We claim that B(3)=P(A(”) and to show this we must verify that each column of B(’) is latin.

Prolongation in m -dimensional permutation cubes

231

All columns of types 1 and 2 in B") are latin. This follows from Lemma 1.1 and the fact that columns remain latin after inverting intercalates. To show that columns of type 3 are latin requires more work. Consider first the columns G;!i2 where 1S i l , izS n. Each C$:!i2 contains exactly one cell of U&,T$i),say (il, iz,j ) , 1 s j s n, and one cell of P(A(')),( i l , i2, n + 1). The remaining n - 1 cells of Cf!i2contain the integers 6i,,b,h= uil,hsi3, i3 # j , all of which are distinct and preclude uil,i,j since A") was a 3-DPC. Two possibilities now arise. If (il, iz,j ) E F3),then 6i1,i2,j = uil,i2,j and 6il.i2,n+~ = n + 1. On the other hand if In either case C$3,is = n + 1 and bil,i2.n+l = ( i l , i2,j ) F3),then bil,i2,n+l latin. Consider now the columns c($ll,i2, 1 s i z S n. The integers in these cells ) so C:?l,i2 is latin. A similar are precisely those of the cells C$:)of P ( A @ ) and argument shows that C$:!n+l, 1 =s il C n, is latin. Finally, we consider C:]I.n+l. The integers in these cells are those in the cells of f13) in A(3)along with n + 1 in t h e cell (n + 1, n + 1, n + 1) and so Cl(n31,n+~is latin. Thus B(3)= P ( A 9 . 0 We remark that for every order n, n # 2, there is a latin square of that order with a transversal. These are easy to construct. For n odd let A(')= (uiI,Jwhere = il + i2, calculations being made mod n on the residues 1,2, . . . , n. This square has transversals T")= {(i, i): 1 S i C n} and S") = {(i, i + 1): 1 5 i s n}. Let P(A('))be obtained using T(*). Then P(A@)) is of even order n + 1 and has a transversal R") = 9 ')u {(n + 1, n + I)}. It is not known if for every n there exists a 3-DPC of order n satisfying the conditions of Lemma 1.2. However, if (n, 6) = 1, n k 5, let A(')= ( u ~ ~ be , ~ a, ~ ) 3-DPC of order n where u ~ , = , ~il,+~ i 2 + i3. All calculations are t o be made mod n on the residues 1,2, . . . , n. Choose TI:)= { ( j , 2(i3- 1) + j ) : 1 s j s a } and F3)= { ( j , 1 - j , 32 - j): 1 S j S n}. Hence A(2) = (ui,,J where uiI,& = iil + sil 1 and T")= { ( j , 1 - j ) : 1S j S n}. This 3-DPC can now be prolongated.

+

2. Prolongation in m dimensions

Before looking at prolongation in m-DPCs let us consider the case when

m = 4. In particular, the CDPC A(4)= ( u ~ ~ . ~of. ~order . , ) n, n 3 7 and (n, 30) = 1,

where uil,i,h,i. = il + i2 + i3+ i4; all calculations being made mod n on the residues 1,2, . . ., n. It is hoped that this example will illuminate both the general prolongation (to be defined inductively), and the observations of Section 3. = (uil,J where Write A(4)as n2 2-DPCs = u ~ , , ~ as , ~ n, ~3-DPCs ; @ = (ki1,h.h) where k-Z1.Q.Q . . = uil,b,h,k; and as n 3-DPCs Pi:) = (pil,h,J where pil.h.&= ~ i ~ . b . h . ~Now, . in each A!& we define a transversal T$:!i4 = { ( j , 2(i3- 1)+ 3(i4- 1)+ j ) : 1 d j S n}. In K$:) we define a transversal S$i)= {(j,2(i4-l)-j+l,2-j-&): l S j C n } and in Pf:) we define a transversal

237

K . Heinrich

R!:) = {(j,2(3 - 2j - 2i3)- j + 1,4 - 2j - 2i3): 1< j < n}. It can be verified that each of K::) and PI:) satisfies the conditions of Lemma 1.2 for prolongation. (Not only this, but if S * = {(il, i2, i3, i4): (il, i2, i 3 ) E Sl,”)} and R * = {(i,, iz. i3, id): (il, i2, id)€ R::)}, then S * = R * . ) Thus we have K f ) = ( k l l . i 2 )where k i , , i 2 = $(il + 3iz - id + 5) and Pj:) = ( p i l , J where pil.t,= J( 2i I + 4i2+ i3+ 5 ) (these are the

A(’) of Lemma 2.1). Finally, we define in A(4) a transversal F4)= { ( j , j , i - ! j , i + j ) : I S j G n } , noting that T(4)CS*. It is now clear that t h e Idimensional arrays K (3 = ) (kil,lZ,y)where ki,.i,y= !(i, + 3iz- i4 f 5) and P(3)= (pil,b,3)where = ;(2iI + 4i2+ i3 + 5) are 3-DPCs and using transversals ‘inherited’ from A(4)(for illustration see Fig. l), also satisfy the conditions of Lemma 1.2 for prolongation. In K$:) we have the transversal S&)= {( j, 2(i4- 1) - j + 1): 1 S j S n}, in Pi:) the transversal R!:)= { ( j , 2(3 - 2j - 2 4 - j + 1): 1 S j s n}, in K(3)the transversal SQ)= { ( j , j, t + j ) : 1 s j 5 n), and in P3)the transversal R(3)= { ( j , j, f 1 S j S n}. In each case, K(3)and P(3),for prolongation we must first define 2-DPCs K(’)and P(2),using the transversals of K$,Z)and P!:)respectively. Doing this we find that K‘” = (kiIJ where k 1l.Q - . = f ( i l + !Tiz + 9) and has transversal S(*)= {(j , j ) : 1 =zj < n}. We also find that K‘” = pea.

3):

In Fig. 1 the 4-DPC A(4),the 3-DPCs K(3)and P3),and the 2-DPC K(’) = Pc2) defined as above are shown in part. The integers in italics show the cells of TC!14,those singly underlined show the cells of S*, S::) and R$:) and those doubly underlined show the cells of F4), S(3),R‘3)and S”). We are now ready to prolongate A(4).First, prolongate each of K$:),Pi:) (according to Lemma 1.2), and Since the transversals S$:) and RI;) were chosen so that S* = R * the prolongations of K$,”)and P!:) are compatible. Consider now the squares P(KI,?)),P(P$:))and P(K(”).These comprise t h e first step in prolongating K(’) and P(3). So we invert the intercalates in each of P(KI,?))and P(P!f’)defined by the cells of S(3)and R(3)respectively; that is, if (il, iz. id) E St” invert the intercalate defined by the cells (il, i2), (i , ,n + I), ( n + 1. i?), ( t i + 1. 11 + 1) in P(K!:)),and if ( i l , i2, i 3 ) € R(3)invert the intercalate defined by that same set of cells in P(P$:)).Thus we have P ( K ( 3 ) )and P(P(3)). However this is not enough. We must now invert all intercalates in Z(P(A$f!i4)) defined by the cells of Fa); that is, if (il, i2, i3, i4)E F4)invert the intercalate of Z(P(A$ZIi4)) consisting of the cells (il, i2), (il, n + l), (n + 1. i2), (n + 1, n + 1). We now have a 4dimensional (n + 1) x (n + 1) X (n + 1) x (n + 1) array B(4) and claim that it is a 4-DPC of order n + 1 obtained by prolongating A“; that is, B(“)= P(A(”).This is not difficult to see. Columns of types 1 and 2 are latin as they simply result from prolongating and inverting (in some cases twice) the latin squares Pj:) and K(’). Columns of type 3 are latin as they are columns of Z(P(Ki:)))and P(P”’); and columns of type 4 are latin as they are columns of Z(P(P!;)))and P(K(3)).

Prolongation in

A (2) 21

A(2) II

4 5 6 7 1 2 3

5 6 7 6 7 1 7 1 2 123 2 3 i 3 4 5 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 1

3 4 5 6 7 1 2

12 A(2) 5 6 7 1 2 3 4 6 7 1 2 3 4 5 7 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 1 3 4 5 6 7 1 2 4 5 6- 7 1 2 3

5 6 7 1 2 3 4

6 7 1 2 3 4 5

4 5 6 7 1 2 3

5 6 7 1 2 3 4

6 7 1 2 3 4 5

7 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 1

p (2) I 4 3 2 1 7 6 5 7 6 5 4 3 2 1 3 2 1 7 6 5 4 6 5 4 3 2 1 7 217:543 -5 4 3 2 1 7 6 1 7 6 5 4 3 2

7 1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 3 4 5 4 5 6 6 7 .*. 6 7 1 7 1 2 1 2 3

S

A$)

2 3 4 5 6 7 1

3 4 5 6 7 1 2

4 5 6 7 1 2 3

4 5 5 6 6 7 7 1 1 2 2 3 3 4

A;) 5 6 6 7 7 1 I 2 2 3 3 4 4 5

7 1 2 3 4 5 6

A? 6 7 1 7 1 2 1 2 3 2 3 4 3 4 5 4 5 6 5 6 7

1 2 3 4 5 6 7 2 3 4 5 6 7 1

2 3 4 5 6 7 1 3 4 5 6 7 1 2

7 1 2 3 4 5 6

1 2 3 4 5 6 7

3 4 5 6 7 1 2

4 5 6 7 1 2 3

5 6 7 1 2 3 4

4 5 6 7 1 2 3

5 6 7 1 2 3 4

A$ 6 7 1 7 1 2 1 2 3 2 3 4 3 4 5 4 5 6 5 6 7

2 3 4 5 6 7 -1

A:) 3 2 3 4 5 6 7 1 4 3 4 5 6 7 1 2 5 4 5 6 7 1 2 3 6 - * . 5 6 7 1 2 3 4 7 6 7 1 2 3 4 5 1 7 1 2 3 4 5 6 2 1 2 3 4 5 6 7

..

Pp’ PF) 6 5 4 3 2 1 7 2 1 7 6 5 4 3 2 1 7 6 5 4 3 j 4 3 2 1 7 6 1 7 6 5 4 3 2 5 4 3 2 1 7 6 4 3 2 1 7 6 5 * * e l 7 6 5 4 3 2 4 3 2 1 7 6 5 7 6 5 4 3 2 1 3 2 1 7 6 5 4 7 6 5 4 3 2 1 6 5 4 3 2 -1 7 3 2 1 7 6 5 i Fig. 1.

233

K(2)

6 7 1 2 3 4 5

A(Z) 17

3 4 5 6 7 1 2

rn -dimensional permutation cubes

I

4 1 5 2 6 3 1

2 7 5 3 1 6 6 4 2 7 5 3 3 1 6 4 2 7 7 5 3 1 6 4 4 2 3 5 3 1 1 6 4 2 7 5 5 3 1 6 4 2

7 41 5 2 6 3

K!’ 5 3 1 6 4 2 2 7 5 3 1 6 6 4 2 7 5 3 3 1 6 4 2 7 7 5 3 1 6 4 4 2 7 j 3 1 1 6 -4275

KY’ 1 6 4 2 7 5 3 5 3 1 6 4 2 7 2 7 5 3 1 6 4 6 4 5 7 5 3 1 3 1 6 4 2 7 5 7 5 3 1 6 4 2 4 2 7 5 3 1- 6 p(2)=

K‘2’

2 5 1 4 7 3 i 7 3 6 2 5 6 5 5 1 4 7 1 4 3 3 6 2 3 6 2 j 1 4 5 1 4 7 3 6 7 3 6 2 5 i

6 1 3 5 7 2 t

234

K. Heinrich

Let us summarize how we have, in effect, obtained P(A(4))from A(4).First, A(4)and its subarrays were required to contain very specific transversals so that each K!:)could be prolongated and we could define another 3-DPC P3)which using transversals inherited from A(4)was also capable of prolongation. Next each of P(K!:))and P(P(’))was constructed and intercalates in P(K&))defined by the cells of T(j)were inverted. (If (il, iz, i3, i4) E T(4),invert the intercalate in P(K$:))consisting of the cells (i,, iz, i3), ( i l , n + 1, i 3 ) , (n + 1, iz, i3), (n + 1, n + 1, i,), (il, i 2 , n + I), (il, n + 1, n + l), ( n + 1, iz, n + 1) and (n + 1. n + 1, n + l).) Due to the particular choice of transversals the resulting array was a 4-DPC of order n + 1. Theorem 2.1. A n

n i -DPC

satisfying cerfain conditions can be prolongated.

Proof. Both the sufficient conditions for and the technique of prolongation will be described inductively. The procedure is in fact a generalization of the Cdimensional case done earlier. The induction is to be done in two parts, the basis of these being prolongation in 2 and 3 dimensions as given in Lemmas 1.1 and 1.2. So throughout assume m 2 4 . First. when a 2-DPC A(’)= (a,J of order n with transversal T(’)is prolongated each ( i t , i z ) E T(’) defines in P(A(*))an intercalate consisting of the cells (i,, i:), ( i l , n + l), (n + 1, i2), ( n + 1, n + 1) and based on the integers and n + 1. This is easily seen from L,emma 1.1. Suppose that when an (rn -2)DPC A(”-*)= .lm-2) of order n is prolongated using the transversal T(m-2). we have in P(A(”-2)), for each (it, i ~ ,. . . , i m - z ) E Fm-’),an intercalate consisting of the cells (il, i ~ ,i?, i ~ ,. . . , in, 2). (i,, n + 1 , iz, i d , . . . , irn-2), (n + 1, i ~ i,, , i4, . . . ,,i z), ( n + 1, n + 1 , i3, i 4 , . . . , ( i l . iz, n + 1, i4, . . . , in,-*), ( i l , n + 1, n + 1 , i4.. . . , i,-?), (n + 1 , n + 1, n + 1, i4. . . . , i m - 2 ) , (il, iz, i3, n + I, ( n + I , i2, n + 1, i4, . . . , i n , - 2 ) , . . . , i,,,-Z), . . . , ( n + 1, n + 1 , n + 1, n + 1 , . . . , n + 1) and based o n the integers all., .In-2 and n + 1, with n + 1 in cell ( n + 1. n + 1, n + 1, n + 1 , . . ., n + 1). We E T(”-’). We now wish to shall say this intercalate is defined by ( i l , i2, . . . , LZ) show that this statement is true in rn - 1 dimensions and to d o this we require another induction. Notice (Lemma 1.2) that in 3 dimensions A(’) is prolongated by first defining in A”’ a transversal F3)and in each of A!:) a transversal 7’;:)so that if (il, iz% i 3 ) E F3’,then (i,, i2) E 7’:;). Next these transversals are used to define A‘” with transversal T(’) where (il. i2)E T(’) if (i,. i2. i3)E T(3).Finally using transversals Tl:) and T”),A!:) and A(*) are prolongated and the intercalates in

Prolongation in m-dimensional permutation cubes

235

P(A!:)) defined by F3) are inverted; that is, if (il, iz, i3)E F3) invert the intercalate in P(A{i))defined by (il, i2) E Tit).We now have P(A"). Suppose now that = (ail,h.,...i,-J and A("-')= (al.i ,.... i,-J where ail,i2,. ..,im-2 . _ . . im-l , are respectively, ( m - 1) and ( m - 2)-DPCs of order n. Suppose also that A$:--:)has transversal Ti:--:) and has transversal F"-') where if (il, i2, . . . , im-l)€ T('"-l),then (il, i2, . . . , im-2)E Ti:::). Finally, suppose that A("-'),using the transversal T("-'),is prolongated by first defining, from transversals in A("-')and its subarrays, an ( m - 2)-DPC A('"-')with transversal T(m-2) = {(il, iz, . . . , im-J: (il, iz, . . . , im-l)E T('"-')}. Then by prolongating each of A!::) and A("-') using the transversals Ti:-;') and T('"-'), and finally if (il, i2, . . . , im-l)E T("-'),then in P(Ai:'";2)) by inverting the intercalate defined by (il, i2, . . . , im-2)E Ti:--:). We now verify the first induction. Consider the cell (il, i2, . . . , i,,-l)E T("-l). Since (il, iz, . . . , im-J E Tit--:) we have in I(P(A$:--:))),by the first induction hypothesis, an intercalate consisting of the cells (il, i2, . . . , im-2), (il, n + 1, ( n + 1, n + 1, . . . , im-z)r. . . , ( n + 1, n + 1, ( n + 1, iz, . . . , im-2), . . . , im-2), . . . , n + 1). In P(A("-'))these are the cells (il, iz, . . . , im-2, im-i), (il, n + 1, . . . , imW2,im-l), ( n + 1, i2, . . . , im+, im-,), ( n + 1, n + 1, . . . , im-2rim-l), . . . ,( n + 1, n + 1, . . . , n + 1, i,,-J, containing the integers a i l , i 2, _ . . ,im_l and n + 1, and, because of inverting, the integer ail.i2 , . . . , im-l is in cell ( n + 1, n + 1,. . . , n + 1, i,,-l). Now, since (il, i2, . . . , i,-2)E T("-'), then in P(A("-')) we have an intercalate based on Uil,Q,...,i m - z , j m ~ l and n + 1, and consisting, in P(A("-')),of the cells (il, i2, . . . , im-2, n + l), (il, n + 1, . . . , im-2, n + l), ( n + 1, i2, . . . , im-z, n + l), ( n + 1, n + 1,. . . , im-2,n + l), . . . , ( n + 1, n + 1 , . . . , n + 1, n f 1) with n + 1 in cell ( n + 1, n + 1, , . . , n + 1, n + 1). Clearly we now have the intercalate in P(A("-'))as required to complete this induction proof. The next step is to verify the second induction. This is more complicated as in doing so we will also be describing the conditions under which an m-DPC can be prolongated. As the reader will see, these conditions can only be described inductively. Let A(")=(u~~,~,....~,,,) be an m-DPC of order n, m 2 4 . Let KI,"-')= ( k i 1 . k ,....i , , , - I ) be the ( m - 1)-DPC of order n defined by k i 1 . i ,... j m - 1 - ai1.i,....i,, Pi:-;').= ( P i 1 . C ....,im-z,im) be the ( m - IkDf'C of order n defined by Pi1.i,...,im-2.im ail,& ,....i,, and A{:--l:]m = (ail,i ,...,im-2) be the (m - 2)-DPC of order n defined by ai1.i. . . . i , - 2 = ail,b, . . . .i,,, for 1s im-ir i m s n . Suppose that A(") and its subarrays contain certain transversals so that each KI,"-')and Pi:-;') can be prolongated according to the induction hypothesis. In particular then, each K$:-') has a transversal S$:-') and has a transversal R!:--l*).We require that if S* = {(il, i2, . . . , im): (il, iz, . . . , i,,,-,) E S$:-')} and R * = {(it, i2, . . . , im): (il, i2, . . . , im-2r i,)E R!:::)}, then S * = R*. Also, in A'")

236

K.Heinrich

we must have a transversal T(")such that Fm)C S*.So, in prolongation we use the transversals of K$r-'),P$:-;') and their subarrays to define ( m - 2)-DPCs K$:--') and PI:--:). Then, because R * = S*, we can simultaneously prolongate K$r--') and P$r:l')by prolongating each of A$Z-;:L,K$:-2)and Pjz--f), and then inverting intercalates in F(A~C--l~~m) defined by the cells of S'; that is, if (it, i2, . , . , im)€ S* invert the intercalate defined by the cell (il, i2, . . . im-z) in P(Aim_l,,m). Notice that in prolongating K!,"-') and P\:--l') we have a fixed set of transversals, so that P(AIZ--:jm)is the same for each prolongation. (Refer back to t h e example at t h e beginning of this section.) This is the first step in prolongating A(m). Now, return to the arrays KIr-.')and P:::,'). Let IQm-')= (kil,i2, _ _i m.-.2 , i m ) and p(m-1)= (Pi1,ip. , i m . , ) where KI:-2' = (ki1.h. . . .i,,-J SO that k i l , . . ,im-2,im - kil.i?... . .i,-z = Pil.h..._.im-?. and PIr--12) = (PiI,t>, . . i m - J so that Pi',?,....i, First, we require that both K("-') and P('"-')be ( m - 1)-DPCs. Notice that in K$:-') we have the transversal {(il, i2, . . . , im-2): (il, i2, . . . , i m - JE S$r-') and in Plr-;;') we have the transversal R$;.;')= { ( i l , i2, . . . , i m - 2 ) : ( i l , i2, . . . irn-', i m )E R$:,')}.Also, we have in ZQm-') and @"-I), respectively, = transversals S("-l)= {(il, i2, . . . , im-z. im): ( i l , i 2 , . . . , i m ) ET'")) and { ( i l . i2, , . . , im-& (i,, iz, . . . , i m )E P"}. We require next that, using these transversals along with others which will have arisen via the induction, both K("-')and P("-') satisfy the conditions for prolongation. In fact. what we want is that in prolongating K('"-')and Pm-'), P(K$,"-2))and P(Piz:;)) are to be exactly the same as they were in the Finally, we require that the ( m - 2)-DPCs prolongation of K $ z - ' )and P$r--l'). defined by the already specified transversals in K("'-')and its subarrays, and by and its subarrays, K("-*) and P(m-2).respectively, be transversals in !'("-I) identical. The transversals in K(m-2) = P'" - 2 ) being defined from K("-') (or equivalently P("-'))via the induction. In particular, the transversal T(m-2) in K("-') is p"-*)= { ( i , . i2, . . . , im-:): ( i l , iz, . . . , i m ) EF'")}. Once all these conditions are satisfied it is not difficult to prolongate A'"'. We simply prolongate each of K$,"-')and P("-')using the transversals Sl2-l)and R("-')(in accord with the induction hypothesis) and then invert intercalates in P(K{:-')) defined by the cells of T'") where if ( i l , i 2 , . . . , i m )E T'") invert the E $:-I). Let us denote the intercalate in P(K$:-'))defined by (il, i2, . . . , resultant array by B(")where BIZ-') = l ( P ( K $ r - ' ) ) ) ,1 S i, S n and BLY;')= P(Pc"-.')).We must verify that B(")is an rn-DPC of order n + 1. Now, because of the requirements imposed on A'"), the same array B'") also arises by prolongating each of P$r--,')and K(m-')and then inverting intercalates in P(P$,"rI')) defined by the cells of F").Consequently all columns in B(")are latin and so B(")= P(A('")).Notice that prolongation in A("' has now been carried o u t in t h e manner described by the induction hypothesis. We simply let

.

.

Prolongation in m -dimensional permutation cubes

237

KI:-') = A!:-') and P("-l)= A("-')with the consistent name change for transversals. The induction proof is now completed. 0 3. Some observations

W e now wish to make some remarks regarding Theorem 2.1. Throughout we shall use the notation of that theorem. It seems at first that the conditions of A'"')are quite unrealistic and that we could rarely hope they would be satisfied. Recall however that they are satisfied in 4 dimensions for order n when (n, 30) = 1. In fact we believe that for infinitely many values of n, where (n, p(m))= 1 €or some p ( m ) , the rn-DPC + i,, calculations made mod n, A(")=( a . . , _ _ ),i _ ,where ,,.., im = i l + iz+ * does satisfy the requirements although this seems to be quite difficult to verify. The following lemma and theorem give us a simple and direct way to construct A("-');this being not immediately apparent in the theorem.

Lemma 3.1. Let A'")= (aii,i,...,im) be an m-DPC satisfying the conditions for prolongation. 7'hen the 2-DPC A!:!i4... ,;, = where = , _ _ im . , has a transversal. Proof. When m = 3 the lemma is clearly true and each A!:) has a transversal (see Lemma 1.2). Suppose the statement of the lemma is true for ( r n - 1)DPCs. Since A(") satisfies the conditions for prolongation, each A!,"-')must also satisfy the conditions and so by the induction hypothesis the result is proven. 0

Theorem 3.2. Let A'")= . . ,Jbe an m-DPC satisfying the conditions for prolongation. Let T\:!.. . ,i, be the transversal in A!:!.. .,im. Then putting B("-')= (hi. . _ .;,-,) . where if ( i l , i 2 ) E Ti:!.. ..i,, hi.....i,-l - ail,...,,i then B(m-1)= A ( m - 1 ) Proof. When m = 3, the statement is seen t o be true by studying the construction of Lemma 1.2. Let us suppose now that the statement is true for (rn - 1)-DPCs.Consider A'") in terms of the (m - 1)-DPCs Pfz:ll)which satisfy the conditions for prolongation. By the induction hypothesis each PjF-;') is constructed from the transversals Ti:!,, . ,i, for fixed i m - l . Consequently so is P("-[); that is, 0 Notice that in order that A("-')exists as an (rn - 1)-DPC constructed from

A'"), it follows from Theorem 3.2 that in A(")we at least have a set of n"-' cells so that each of the integers 1,2, . . . , n occurs nm-' times in these cells and

238

K . Heinrich

in every column of A(") exactly one cell is chosen (although this is not sufficient). Beljavskaja and Marathudjaev [ 11 have called such sets diagonals and have made some studies of them. Finally, let us say that we do not believe all the requirements of Theorem 2.1 need be explicitly stated. In particular, given the necessary requirements on K$E-').P!;:,') and K("-') we believe that the conditions on P("-') and K("'-*)= P("'-') will follow. However, a verification of this fact may not justify the work involved.

Acknowledgement The author wishes to thank the Mathematics Departments at the University of Newcastle and the University of Queensland for their hospitality and assistance during the summer of 1980.

Note added in proof The author has recently learnt that Frank Walther (in: Wissenschaftliche Zeitschrift der Padagogischen Hochschule) also has studied this problem and other, closely related to it. In particular, he has proven Lemma 1.2. Simon Fraser University Burnaby Vancouver, Canada

References f I] G.B. Beljavskaja and S. Murathudjaev, About admissibility of n-ary quasigroups. Com-

binatorics, Colloq. Math. Jdnos Bolyai 18 (1978) 101-1 19. [2] V.D. Belousov, Extensions of quasigroups, Bull. Akad. Stince RSS Moldoven 8 (1967) 3-24: MR 38 (1969)4592. 131 R.H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1%) 245-354. [?I J. D i n e s and A.D. Keedwell, Latin squares and their applications, Akademiai Kiad6, Budapest, 1974.

Annals of Discrete Mathematics 15 (1982) 23%251 @ North-Holland Publishing Company

MATCH-TABLES A.J.W. HILTON and C.A. RODGER Dedicated io N.S.Mendekohn on the occasion of his 65th birthday In this paper, the concept of a match-table is introduced. W e may think of a match-table as a season's fixure list for a group of teams, where the games are not part of a competition and so are only played for enjoyment. Teams may play each other several times or even not at all, the main requirement being that everyone plays as often as possible. To construct such a match-table, a good first step might seem t o b e t o construct an outline match-table, where perhaps all teams from towns that are geographically close or all teams of similar abilities for instance are grouped together. Having constructed an outline match-table satisfying the main outline requirements, one might then develop this into a complete match-table. This paper shows that for the appropriate kind of outline matchtable, this is always a feasible approach. Furthermore. conditions are given t o ensure the completion of a partial match-table into a match-table without altering any of the partial match-table.

1. Introduction

Let M be a symmetric n X n matrix o n b symbols T ~.,. . , 76 with each cell containing any number, possibly zero, of symbols and possibly with repetitions of a symbol within a cell. Then M is a match-table if the following conditions are obeyed for 1 j S b: (i) for each symbol q,there exists an integer f, such that T~occurs exactly ti times in every row and column of M ; (ii) if nQ is even, then rj never occurs in a diagonal cell of M; (iii) if n4 is odd, then q occurs in exactly one diagonal cell of M ; (iv) [ x i- &I S 1 for 1 C i, k S n where xi is the number of symbols occurring in M i . i . Define r, to be the frequency of q in M. We may think of the rows and columns as representing cricket teams involved in friendly weekend 'matches, where no winner of a competition is sought and so it becomes unnecessary for each team to play each other the same number of times. Then the symbols in Mik represent the weeks of the season that team i plays team k. Thus normally one would have tk = 1, but if perhaps a second team can be raised occasionally by each club, on bankholidays for example, then this can be represented by setting ti = 1 for most of 239

240

A.1.W. Hilton, C.A. Rodger

the weeks 7, and rj = 2 for the weeks in which two teams can be raised. The usual process for organising matches may simply involve the club secretaries writing to the clubs they wish to play against and suggesting possible dates. By such a hit and miss method, a satisfactory match-table is usually achieved. However, a team with a relatively inactive secretary may find itself with several matchless weekends. If, instead, a letter detailing preferable dates and matches were sent by each club secretary to an area secretary who could co-ordinate matters he might well consider using the process described in this paper to obtain a satisfactory match-table. In such a competition it may be common for stronger teams, sociable teams or geographically close teams to want to play each other comparatively often, and such prerequisites can be incorporated into the match-table. We presume that each team wants to play as often as possible, and so the number of byes, represented by symbols on the diagonal of M,is kept to a minimum; the byes are spread as evenly as possible among all the teams. In Section 2 of this paper we discuss the problem of constructing, or reconstructing, match-tables. Essentially, what we do is to form first an ‘outline match-table’ which may incorporate various requirements or prerequisites and which may be considerably simpler, and therefore easier to construct, than a match-table. From this we show how to evolve a match-table. In Section 3 we show exactly when some partial match-tables can be completed to form proper match-tables. We obtain necessary and sufficient conditions, reminiscent of those of Ryser [8] for the completion of latin squares. We should remark that the results here are analogous to the results in [7], but that here the matrices are symmetric. Symmetry makes the arguments more difficult and restricts somewhat the results which are obtainable. Finally, in our definition of a rnatch-table we permitted repetition of a symbol in a cell. Although we have not worked the details out, there is every reason to suppose that similar results to ours hold when repetition is not permitted (see, for example, [l,21). Similarly if we were to drop condition (iv) from the definition of a match-table, we are confident that theorems very similar to ours would still be true.

2. Reconstructing match-tables 2.1. The reduction modulo

(P,P, S ) of a match-table M

A composition C of a positive integer n is a sequence (cl, . . . ,c,) of positive integers such that c1+ c2+ * + c, = n.

--

Match-tables

241

Let an n x n match-table M on the symbols 1 , . . . , b be given and let P = (PI,.. . , p n ) and S = ( ~ 1 , . . . , s,) be two given compositions of n and b respectively. We now define the reduction modulo (P, P, S ) of M as follows. Intuitively it is t h e matrix obtained from M by amalgamating rows p I + . .+ + 1,. . . ,pl + * + p i , columns pl + * + pi-l + 1 , . . . ,p1+ * + p i and symbols S 1 + * * . + S k - l + 1 , . . . , S l + . * . + S k for 1 S i G U and 1 S k S W . More precisely, for 1 S A, p G u and 1 S 6 S b, let x(A, p, 5) be the number of times that symbol 6 occurs in the set of cells

-

and, for 1 =Sv

S w,

let

%”(A, p ) = x(A, p, S I + . . . + S,-I

+ 1) +

* * *

+ x(A,

p, SI+ *

. + s,) . *

Then t h e reduction modulo (P, P, S ) of M is a u x u matrix whose cells are filled from a set of w symbols (say {q,. . . , T , } ) and in which cell (A, p ) contains T,x,,(A, p ) times. Given a reduction modulo (P, P, S ) of a match-table M, clearly this itself may be further reduced by further amalgamations of rows and columns or of symbols. We now illustrate this reduction process with an example. Let the given match-table be as in Fig. 1. Then n = 9 and b = 6. Let u and w both be 5 , and let P = (1,1,2,2,3) and S = (1, 1, 1,2, 1). The frequencies of the symbols 1, 2, 3, 4, 5 and 6 are 1, 1, 2, 2, 2 and 3 respectively. As only t l , t2 and f6 are odd, only the symbols 1, 2 and 6 occur on the diagonal of M, and occur exactly once. The symbols occurring in diagonal cells are shared as equally as possible between the diagonal cells. Let I be the composition of a sequence (of length appropriate to the context) of 1’s. Then the reduction modulo (I, I, S ) of M is given in Fig. 2, where the symbols 4 and 5 have been replaced by the symbol a. In this diagram and also in Figs. 1 and 3, there is not intended to be any significance in the way the symbols are arranged in each cell. Finally we reduce modulo (P, P, I ) and obtain the diagram shown in Fig. 3.

2.2. Outline match-tables In this subsection we define an outline match-table, and it is clear (see Proposition 1, below) that a match-table, amalgamated in the way described above, is an outline match-table. We show in Theorem 3 that each outline match-table is the reduction modulo (P, P, I ) of a match-table for some

A.J. W.Hilron, C.A. Rodger

242

6

Fig. 1.

Fig. 2.

Match -tables 2

243 3 a a a 6

6

2 3 a a a

5 6 5

1 % 2 U r I S ! i

3

3 a a a

1 1 3 3 3 3

1

a a n a 6 E

Fig. 3.

composition P, and in Theorem 6 that some outline match-tables are the reductions modulo (I, I, S) of match-tables for appropriate compositions S. In this paper, let [xl represent the least integer greater than x and let 1x1 represent the greatest integer less than x. We now define an outline match-table. Let C be a symmetric u x u matrix on b symbols q , .. . , 76,with each cell containing any number, possibly zero, of symbols and possibly with repetitions of symbols within a cell. Let n be a positive integer. For 1 d A =z u and 1 d v G b, let pA and t, be positive integers such that the number of symbols, including repetitions, occumng in row A is p,,, and such that the symbol 7,occurs nt, times in C. Let s = Zt=, t,. Then C is an outline match-table if the following properties are obeyed for each A and v such that 1 c A s u and l c v s b : (i) s divides pA; (ii) the number of times 7, appears in row A is ( p J s ) t v ; (iii) if nt, is even, then 7, occurs an even number of times in each diagonal cell of C;if nt, is odd, then 7, occurs an odd number of times in exactly one diagonal cell of C ;

A.J.W.Hilton, C.A.Rodger

24.4

(iv) if pJs = 1 , then 7” occurs at most once in C A , A ; (v) if C,, contains x, symbols occurring an odd number of times, then @Js) Ld/nJ s X, G @ J s ) [ d n l , where d is the number of symbols for which nt, is odd. Proposition 1. For any composition P, of n, the reduction modulo (P, P, I ) of a match-table is an outline match-table and has the further properties: (vi) @ I . . . . , p I ) = (SPI.. . . vC); (vii) X i r 1pA = ns.

.

Proposition 1 is easy

to

verify.

2.3. Forming match-tables from outline match-tables The graph theory terminology we employ here is standard if it is used without explanation, and may be found in [5] or [ 131. Let G be a graph with vertex set V and edge set E. Let G contain multiple edges but no loops. An edge-colouring of G with colours 1,. . . , k is a partition of E into k mutually disjoint subsets C,, . . . , Ck.Thus Cl n C, = 0 (1 S i < j S k ) and CIU . . * U Ck= E. An edge has colour i if it belongs to C,. Note that we do not make the usual requirement that two edges having the same colour do not have a vertex in common. Given an edge-colouring of G, for each u E V let C , ( u ) be the set of edges on u of colour i, and, for each u, u E V, u # v let Ci(u, v ) be set of edges joining u to u of colour i. An edge-colouring of G is called equitable if, for all v E V, (a) maxIr;<,6k I IC,(v)l- IC,IIs 1, and it is called balanced if, in addition, for all u, u E V, u f u, fb) maxlc;<,lcltI IC,(u,u)I - IC,(u, v)l I 1. Thus an edge-colouring is balanced if the colours occur as uniformly as possible at each vertex and if the colours are shared out as uniformly as possible on each multiple edge. De Werra [lo-121 proved the following important theorem.

-ition 2 (de Werra). For each k edge-colouring with k colours.

L 1, any finite bipartite graph has a

balanced

We now make use of de Werra’s theorem to form match-tables from outline match-tables.

Theorem 3. To each outline match-table C, there is a match-table M and composition P such that C is the reduction of M modulo (P, P, I ) .

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245

Proof. Suppose that u < n . Then we show that C can be obtained from a ( u + 1)x ( u + 1) outline match-table by the amalgamation of the cells of two rows and columns. Repeated applications of this will show that C can be obtained from an n x n outline match-table, i.e., a match-table. Since u < n, and s divides p l , . . . , po, and CX=IpAls = n, there is at least one A for which pJs > 1. Without loss of generality we may assume that pJs 3 2. We wish to form an outline match-table C' by splitting the last row and column of C into two new rows and columns. To do this, we construct a bipartite graph G with vertex sets (yl, . . . , and ( T ~.,. . , T,,), where, for 1 s i s u - 1 E", and 1 S j S b, yi is joined by exactly k edges to the vertex T~ if and only if the symbol 7 occurs k times in the cell C,,i.Join vertex T~ to E" with one edge if and only if T~ occurs an odd number of times in C,,and join T, to with 2Ltk'J edges if and only if symbol 7j occurs k' times in Cu.,. Clearly the degree of 7 in G is (pJs)tj for each j , and the degree, d ( ~ " ) o,f E" satisfies (PJs)Ldlnl s d ( E " ) s (pJs) [dlnl. Now give a balanced edge colouring to G with pJs colours, say 1,2, . . . , pJs. Then each vertex T~ has exactly ti edges of each colour on it. We now split row u of C into two rows u * and u * * . For I S p S u - 1, place symbol in cell ( u * , p ) if vertices q and yp are joined by an edge coloured 1, and in ( u * * , p ) if they are joined by an edge of any other colour. Also split column u into two columns, and, for 1 S i S u - 1, fill the cell (i, u * ) and (i, u * * ) with symbols such that the matrix remains symmetric. Then it remains to divide C , , into four cells. If has 2kj edges joined to 7j of which k;' are coloured with 1, then place symbol T~k l times in both the cells ( u * , u * * ) and ( u * * , u * ) , and 2(kj - k l ) times in cell ( u * * , u * * ) . The balanced colouring of t h e multiple edges joined to E , ensures that kj- k ; ' 0. ~ Finally, for each vertex T~ joined to E", if the connecting edge is coloured 1, place symbol T~ in cell ( u * , u * ) , and otherwise in cell ( u * * , u * * ) (i.e., place T~ an additional time in ( u * * , u * * ) if it is already there). We can now check that the ( u + 1)x ( u + 1) matrix C' formed by the rows and columns 1,2, . . . , u - 1, u * , u * * forms an outline match-table. Let p; be the number of symbols in row A of C' and let x i be the number of symbols that occur an odd number of times in C;,A.As each vertex T~has exactly rj edges on it that are coloured 1 in G, we know that p:* = s, pL.0 = p, - s and that pi = pi for 1 c i C u - 1. Also, each symbol q occurs exactly ti times in row u* of C' and exactly (pJs - l)tj times in row u * * of C', and so C' satisfies conditions (i) and (ii) of an outline match-table. By the construction of G, a symbol that occurs an even number of times in C,,also occurs an even number of times in both C:.,,. and C',*O,u-, and a symbol occurring an odd number of times in C,, occurs an odd number of times in exactly one of C,*,u-and Cuo*,u-. Clearly contains each symbol at most once. Also, if pJs - 1 = 1, then by the balanced colouring of G, kj- ky = 0 for 1d j d b and so CU-.,,contains each symbol at most once. Therefore C' satisfies conditions (iii) and (iv) of an

A.J.W.Hilton, C.A. Rodger

246

outline match-table. Since E , ) has degree between ( p J s ) [ d / n ] and ( p J s ) [ d / n l and since G was given a balanced colouring with pJs colours, there are either [din] or [d/nl symbols in C,.,,.. Suppose that d = an + c, where 0 C c C n - 1. Clearly, if x, = (pJs)Ld/nJ,then x:. = Ld/nJ and so x l - = (pJs - l ) [ d / n ] ,and if x, = @ J s ) [ d / n 1 , then xb. = [ d / n l and so x;.. = (pJs - l ) [ d / n l . For any other value of xu, we have (pJs)a + I =z x,

c (p,Js)(a+ 1) - I , (pJs)a+ l - ( a + l ) ~ x ~ . - ~ ( p J s ) ( a + I ) - 1 - a , or, in other words,

(p:../s)a s x:.. c (pk../s)(a+ I) or

(p:..ls)Ld/n]

X:-C

(p:-/s)[d/nl .

Therefore both Ci*.v-and Cv-.,-satisfy condition (v), and as Ci,i= C,.ifor 1 S i 6 u - 1, C' is a ( u + 1) X ( u + 1) outline match-table. By iterating this procedure, we obtain an n x n outline match-table o n b symbols. This proves Theorern 3.

Corollary 4. Let M be an n X n match-table on the symbols 7 1 , . . . , q, with associated frequencies I , , . . . , tb. Then, for every integer x that divides t, for each 1 S j =s b, there exists an (nx) X ( n x ) match-table M' on the symbols 71, . . . 7 6 with associated frequencies tJx, . . . , tdx and a composition P = (x, x. . . . , x ) such that M is the reduction of M' modulo (P, P,I ) .

.

Theorem 3 may be of some practical use when drawing up a match-table. One might, for example, as a first step, draw up an outline match-table in which teams situated close together, teams which are of a different standard to the rest of the teams, or teams that do not wish to play against each other are each grouped together. Symbols, representing the week of the season when two teams are to meet, can then be placed in the outline match-table to satisfy requirements. For instance, t h e first group mentioned above may wish to play each other fairly often to save travelling costs, and so the corresponding diagonal would contain many symbols, whereas the diagonal of the third group would be empty. When this outline match-table is deemed to be suitable, it might then be developed into a match-table. The proof of Theorem 3 uses the fact that the edges incident with the symbol 7, are given a balanced colouring. but does not use the balanced colouring given to edges incident with the vertex y, or The effect of this balanced colouring is that, when a cell ( i , j ) of C divides into several cells, the symbols in

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247

(i, j ) are spread as evenly as possible amongst the divided cells, allowing of course for the diagonal conditions to be maintained first. This is an important feature of the process of forming a match-table from an outline match-table. It means that, given the initial requirements made in the outline match-table, each team plays each other team as few times as possible, and so meets a vaned opposition. The match-table in Fig. 2 is obtained from the outline match-table in Fig. 3 by the process described in Theorem 3, so this evening process can be clearly seen.

2.4. Operations on the symbok In Theorem 6 we see that in certain cases, we can do a similar thing to the symbols as we have just done in Theorem 3 to the rows and columns. This could be used to group months together, to group full day games separately from half day games, or to. differentiate between Saturday and Sunday games. Lemma 5 (Petersen). A regular multigraph of even degree 2m has m edge-disjoint

2-factors.

==

Theorem 6. To each n x n match-table M with 4 = 2m for some j , 1 d j b, there is an n X n match-table M ’ in which m symbols, say T;,. . . , r k , each have frequency 2, such that M is the reduction of M’ modulo (I, I, S ) where S = (1,1, . . . , 1, m, 1, . . . , 1, l), the m occum’ng in the j-th coordinate. Proof. Suppose symbol q has frequency 2m in the match-table M. Then form a graph G with n vertices by joining, for 1d i C k d n vertex i to vertex k by as many edges as the number of times that T~ occurs in M i k . Since ntj is even, q does not occur in a diagonal cell of M and, as M is symmetric, we need only consider i C k. Then G is a regular graph of degree 2m, and so by Lemma 5 , is 2-factorable. Colour the m 2-factors of G with colours 1,2, . . . ,m respectively. Then for 1d v d m, form the match-table M’ by replacing symbol T~ in both Mi,k and M k , iwith the symbol r;” if and only if vertices i and k are joined by an edge coloured with v. Therefore, as every vertex of G is contained in every 2-factor, each symbol 7.; occurs twice in every row and column of M’. It is clear that M’ is a match-table, and by relabelling the symbols of M’ so that TI!. becomes 71,it can be seen that M is the reduction of M’ modulo (I, I, S). This proves Theorem 6.

A.J. W.Hilton, C.A. Rodger

2.M

Corollary 7. If M is a match-table in which the frequency of every symbol is even, then M is the reduction of a match-table in which the frequency of every svmbol is 2 . More generally, if we allow ti to be either even or odd, then we can form the graph G from t h e symbol q as described in Theorem 6, where G contains one loop when nt, is odd. Then it is worth noting that if G has a 1-factor for some 1 < r,, where a loop is regarded as one edge and contributes one to the degree of the vertex with which it is incident, then T, can be replaced by two symbols with frequencies 1 and t, - I so that a match-table results. It is also worth noting that, if there are n o repetitions in any of the cells, then, by Vizing's theorem [9], since G is regular of degree 4, G can be edge-coloured with t, + 1 colours. So we could replace the symbol T~ in the cell (i. k ) of M with t h e symbol T~~if the edge ( i , k ) in G is coloured with I. Each vertex of G would then be missing exactly one of t h e tj + I colours, and so the symbol corresponding to the missing colour would have to be put in the diagonal cell. This will not result in a match-table because at least one of the new symbols occurs in more then one diagonal cell. However, this table could still be useful. For example, consider a convention of many people, where each person wants to speak to several other people, but always in pairs. An outline match-table could be drawn up where the symbol T~in M j , k means that i would talk with k sometime in t h e jth session of the conference. Once this had been done. the jth session could then be divided into t, + 1 periods in the manner described. This would mean that every person would be talking to somebody for t, of the r, + 1 times in each session, and would probably be glad of the rest in the other period.

3. Partial match-tables

An r x r partial match-table on b symbols is a symmetric r x r matrix R in which each cell contains a number, possibly zero, of symbols, which may be repeated within a cell, drawn from the set (7,. . . . ,T*}of b symbols, such that the following is true. Each symbol occurs at most once on the diagonal of R, and there exists at least one pair (d. z ) of integers such that

and

z is odd

if d > 0

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249

where xi is the number of symbols in Ri,i(1 S i S r). When there is more than one such pair ( d , z ) of integers then we differentiate between these by saying that the partial match-table has parameters d and z. The following theorem shows when a partial match-table can be built into a match-table without altering or adding any symbols to R. It is clearly of use when trying to decide whether a match-table can be formed satisfying certain prerequisites. It may be desirable to give certain teams a bye, as they may know that they are unable to raise enough players on certain days, and so this constraint can be included in R. However, most teams would still want as few byes as possible, which corresponds to choosing d to be as small as possible.

Theorem 8. A partial match-table R of size r X r on the symbols rl, . . . , 76 with parameters d and n can be completed to an n x n match-table M on the same symbols with given frequencies t l , . . . , tb of which d are odd and in such a way that no further symbols are placed in any cell of R if and only if the following conditions are obeyed: (i) N ( j ) a ( 2 r - n)4 ( I S j s b); (ii) rj s ti ( 1 s j s b); (iii) if n = r + 1, then N U ) s ( n - 2)tj + 1; (iv) ti is odd for all symbols rj that occur on the diagonal of R ; (v) d - ( n - r)rd/n] s EL1xi S d - ( n - r)Ld/nJ, where N ( j ) denotes the number of times 7j. occurs in R, rj is the maximum number of times 5 occurs in any row or column of R and xi is the number of symbols in Ri.i.

Proof. Necessity. Suppose that R can be completed to form M with n o further symbols being placed in any cell of R, and suppose that A4 is subdivided as indicated in Fig. 4.

-"Fig. 4.

7-50

A.J. W. Hilton. C.A. Rodger

Any symbol r, occurs ( n - r)f, times in B, at most ( n - r)t, times in A, and since T, occurs nt, times in M , it must occur at least nt, - 2f,(n - r ) = (2r - n)f, time!, in R. Clearly. as 7,occurs f, times in every row and column of M, r, f,. By the definition of a match-table, a symbol T, occurs exactly once in a diagonal cell if nt, is odd, and no times if nf, is even. Therefore I, must be odd for each symbol that occurs on the diagonal of R. By condition (v) in the definition of a match-table, [ d / n J c x, S [ d / n l for 1 s I s n. and s o ( n - r ) Ld/n] s X;=,+,x, =s (n - r ) [ d / n ] .However, since nf, is odd for exactly d symbols. E;=, x, = d, and so

Finally, when n = r + 1, since the symbol T, occurs f, times in the final row of M and in the final column and at most once in M,,.,,, we have N U ) = ( n - 2)4 o r (n - 2)f, + 1. Sufficiency. Let R be given satisfying conditions (i)-(v). Form a new matrix R * by adjoining one further row and column to R as follows: for 1 S m S r and I S j s b, cells ( m , r + 1) and ( r + 1, m ) contain T, enough times that it occurs exactly 1, times in row m and column m respectively; this is possible since r, S f, (1 S j s b). Next, for 1 < j < b, symbol T, is placed in cell ( r + 1, r + 1) of R * the number of times necessary that 7, may occur ( n - r)f, times altogether in row r + 1 of R*. Since N ( j ) a ( 2 r - n)f,, the cells in the first r columns of row r + 1 contain rf, - N ( j ) C (n - r)r, entries, so the manoeuvre just described is possible. We now show that R * is an outline match-table. It is easy to see that conditions (i) and (ii) for an outline match-table are satisfied. If n = r + 1, then for R * to satisfy conditions (iii) and (iv) of an outline match-table we require that. for each j , 1 j < b, the number of times 7,occurs in R;," is zero if nf, is even and at most one if nr, is odd. By the construction of R * , 7,occurs ( n - l)t, times in the first n - 1 rows of R* and f, times in the nth column of R*, and so condition (iii) of the theorem ensures that R:" does satisfy these conditions of an outline match-table when n = r + 1. Now consider n > r + 1. Clearly, R* satisfies condition (iv) for an outline match-table. If nf, is even, 7,does not occur on the diagonal of R and so, since R * is symmetric, 7,occurs an even number of times in cell ( r + 1, r + 1) of R*. If nf, is odd, since 7,occurs at most once o n the diagonal of R and since R" is symmetric, 7,occurs an odd number of times in exactly one diagonal cell of R*. Therefore, R * satisfies condition (iii) of an outline match-table. Let x : be the number of symbols which appear an odd number of times in cell (i, i) of R*. Then

March-tables

251

and so we only need to check that

to ensure that R * is an outline match-table. Since each of the d symbols with odd frequency occurs an odd number of times in exactly one diagonal cell of R*, it follows that X;L\ x: = d. Combining this with condition (v) of the hypotheses proves that R * satisfies condition (v) of an outline match-table. Therefore R* is an outline match-table, so the sufficiency follows from Theorem 3. This completes the proof of Theorem 8. Acknowledgement

’

W e would like t o thank the referee for his very helpful and perceptive remarks, as a result of which some important changes were made. Department of Mathematics University of Reading Whiteknights Reading RG6 2AX, England

References [l] L.D. Andersen and A.J.W. Hilton, Generalized latin rectangles I: construction and decomposition, Discrete Math. 31 (1980) 125-152. [2] L.D. Andersen and A.J.W. Hilton, Generalized latin rectangles 11: embedding, Discrete Math. 31 (1980) 235-260. 131 L.D. Andersen and A.J.W. Hilton, Generalized latin rectangles, Proc. One-day Conf. Combinatorics at the Open University, in: R.J. Wilson, ed., Research Notes in Mathematics 34 (1978) pp. 1-17. [4] L.D. Andersen and A.J.W. Hilton, Quelques t h t o r h e s sur carrts latin generalists (oh sur graphes complets tquitablement colorts), Coll. Math. Discrktes: Codes et hypergraphes, Cahiers Centre etudes Rech. Op6r. m(3.4) (1978) 307-313. [5] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [6] A.J.W. Hilton, The reconstruction of latin squares with applications to school timetabling and to experimental design, Proc. CW9 at the University of East Anglia, Math. Programming Stud. 13 (1980) 68-77. [7] A.J.W. Hilton, School timetables, to appear. [8] H.J. Ryser, Combinatorial Mathematics, Carus Mathematical Monographs (Wiley, New York, 1%3). [9] V.G.Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz 3 (1964) 25-30. [lo] D. de Werra, Balanced schedules, INFOR 9 (1971) 230-237. [ l l ] D. de Werra, A few remarks on chromatic scheduling, in: B. Roy, ed., Combinatorial Programming: Methods and Applications (Reidel, Dordrecht, 1975) pp. 337-342. [12] D. de Werra, O n a particular conference scheduling problem, INFOR 13 (1975) 308-315. [13] R.J. Wilson, Introduction to Graph Theory (Oliver and Boyd, Edinburgh, 1972).

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Annals of Discrete Mathematics 15 (1982) 253-258 @ North-Holland Publishing Company

ON THE SEQUENCEABILITY OF DIHEDRAL GROUPS

G.B. HOGHTON and A.D. KEEDWELL Dedicated to N.S. Mendelsohn on rhe occasion of his 65th birthday In previously published papers Friedlander has shown that the dihedral group Dp of order Zp is sequenceable for all primes p = 1 mod 4 and Anderson has shown that Dp is sequenceable for primes p which have a primitive root 7 such that 37 = -1 m o d p provided that a certain design involving a subset of the integer residues m o d p can be constructed. Here we show that Dp is sequenceable for a further infinite class of primes p ; namely for all primes p 9 7 mod 8 for which 2 has the exponent f ( p - 1). We also provide strong evidence for the conjecture that the dihedral group D" of order 2 n is sequenceable for all odd integers n. In particular, we verify that this is the case for all odd n in the range 5 s n =s35.

1. Introduction

A finite group (G, of order n is said t o be sequenceable if its elements can be arranged in a sequence a. = e, al,a2,. . . , a,-l in such a way that the partial products ao,aoal, w z l a 2 , . . . , aoalaz * * are all distinct (and consequently are the elements of G in a new order). It was shown by Gordon 131 that a finite abelian group is sequenceable if and only if it is a direct product A x B, where A is cyclic of order 2" (k > 0) and B is of odd order. It is also known that every countably-infinite group is sequenceable [ 5 ] , that the non-abelian groups of order pq, where p is an odd prime which has 2 as a primitive root and q = 1 + 2ph is any other prime greater than p for which such a group exists, are sequenceable [4]and that certain of the dihedral groups are sequenceable. As regards the latter, Friedlander [2] has shown that the dihedral group Dp of order 2p is sequenceable for all primes p = 1 mod 4 and Anderson [l] has shown that Dpis sequenceable for primes p which have a primitive root 7 such that 37 = - 1 mod p provided that, for the prime p in question, a certain type of design involving a subset of the integer residues mod p can be constructed. We shall show here that Dp is sequenceable for a further infinite class of primes p ; namely for all primes p = 7 mod 8 for which 2 has the exponent i ( p - 1) or, equivalently, primes p = 7 mod 8 which have a primitive root u such that 2u -1 mod p. We shall show, more generally, that the dihedral group 0, of order 2n is sequenceable for any odd integer n for which a particular type of design involving the integer residues mod n can be constructed and verify that such designs exist for all odd integers n in the range 5 d n =s35. 0

)

253

2.54

G.B. Hoghbn, A.D. Keedwell

2. Preliminary results Definition. If a is any sequence uI, u 2 . .. . , u,, of elements of a group G, we shall denote by P ( a ) its sequence of partial products u1 = u I , u2 = u1u2,u3 = uIu2u3,.. . , u,, = ~ 1 ~ *2 4. ~ 3 Let D. be the dihedral group of order 2n and let H be its unique cyclic normal subgroup of order n. Let D J H = gp(l,x}. where x ' = 1. A sequence a of length 2 n consisting of elements of DJH is called a quotient sequencing of D,, if each of the elements 1and x occurs n times in both a and P ( a ) . The image under the mapping q:D,,+DJH of a sequencing of D,, is However, it is not true that every quotient always a quotient sequencing of 0,. sequencing arises in this way (see [2]). It is easy to check that a = (1. x, 1, x, 1, x, . . . , 1, x} is a quotient sequencing o f D , , a n d t h a t P ( a ) = { l . x , x , 1 , 1 , x , x . 1, . . . , l , x , x , l , l , x } i f n i s c .d. Let us assume that a arises from a sequencing 2 of D,, and that the sequence P ( 2 ) is then e, b, baa],a"1, a-, bah, bah, a"', a"4, ba04, buB5, a"', . . . , a " n - l . b a P n - l . in that case, the sequencing a must be as follows: e, b, aal, baQl+"l, aq-"l, baBZ+"Z,a&-&, baP~"3,aa4+n3 ba84+"4,aSsrP4, ba05+"5,. . . , a an- I - ""-2 ba 8. - I+""- I We conclude that a sequencing of the above form exists if the following conditions can be met: (i) the elements 0, al,a2,. . . , a,,-l are the distinct residues mod n ; (ii) the elements 0, PI,P2, . . . , are the distinct residues mod n ; (iii) the elements 0, PI+ a,, P?+ a2,. . . , + a,,-l are the distinct residues mod n ; (iv) the elements 0, P, a2- a ~ .P3- P2, a4- a3, P5- P4. . . . , P n 2 - P n - 3 3 a n e-lan.?are t h e distinct residues mod n. Suppose that a, = mP, and that both m and m + 1 are relatively prime to n. Then, if condition (ii) holds so also d o conditions (i) and (iii) and the sequence given in condition (iv) becomes 0, PI, m (P2 - PI), P3 - P2, m(P4- &), P5 - P 4 , . . . ,& - P,, 3. m(Pn-l- on.?). Consequently, we have the following theorem.

Theorem 1. A sufficient condition for the dihedral group D,,, n odd, to be sequenceable is that there exist an ordering of the complete set 0, PI.P2,. . . , of integer residues mod n together with a suitable multiplier m such that the elements 0, P I . m ( P z - P I ) , P3-Pz. m ( p 4 - p 3 ) , . . . , P n - 2 - P n - - 3 , m(Pn-i-Pn-2) are uguin the integer residues mod n and such that both m and m + 1 are relatiuely prime to n. We note that the values -2 and i(n - 1) for m always satisfy the requirement

On the sequenceability of dihedral groups

255

that both m and m + 1 be relatively prime to n. However, we observe also that, because the cyclic group C, of odd order n is not sequenceable (by Gordon's theorem), the value m = 1 can never satisfy the requirements of Theorem 1.

3. Special cases We consider first the special case when n is prime, n = p say, and let u be a primitive root of p. In this special case, m and m + 1 are relatively prime to p for all choices of m, O C m < p - 1. We take 0, 1, -1, u, -a, uz, as the sequence 0, PI,P 2 , .. . , Then we -uz,. . . , . . . , u(n-3P , + l),-2m, require that the elements 1, u + 1, u(u+ l), u2(u+ l ) ,. . . , u("-'y2(u -2mu, -2mu2,. . . , -2mu(n-3~2 be all distinct modp. This will be the case if 1= + 1) and -2m = U ( " - ' ~ ( U +1). That is, if 2 u = -1 mod p and 2m = u + 1 mod p. However, we require m f p - 1 mod p. If m = p - 1, then -2 = u + 1, whence u = -3. Then, since 2 a = -1 mod p, we have -5 = 0 mod p and so p = 5 . We deduce that the following theorem holds.

Theorem 2. If p is a prime ( p # 5 ) for which the solution of the equation 2 u = -1 mod p is a primitive root, then the dihedral group Dp of order 2p is sequenceable. We require to know which primes p have this property.

Theorem 3. If p is a prime for which the solution of the equation 2 u = - 1 mod p is a primitive root, then either (i) 2 is a primitive root mod p and p = 5 mod 8, or (ii) 2 has the exponent f ( p- 1) mod p and p = 7 mod 8. Proof. Since (-2)u = 1 mod p, it follows that u is a primitive root mod p if and only if -2 is a primitive root mod p. Suppose that g is some _. primitive element m o d p and that 2 = g". Then -2 = g(~-1)R+a,Since -1 +,.g(p-'YZ. -2 is primitive mod p provided that f ( p- 1 ) + a is relatively prime to p - 1. Let p - 1 = 2'"q61q9' * q?, where the qi are distinct odd primes. Then $ ( p - I ) + CT = Zm-lqPq$? * q? + CT. There are now two cases to consider. Case 1. m > 1. In this case, -2 is primitive if and only if a is odd and has none of the qi as prime factors. When a is odd and has none of the qi as prime factors, it is relatively prime to p - 1 and so 2 = g" is primitive. Moreover, m > 1 implies p = 1 mod 4 and a odd implies that 2 = g" is a quadratic = (-l)(+lp whence 2 is a quadratic residue non-residue mod p . We have when p = a l mod8 and a quadratic non-residue when p = a 3 m o d 8 . Since

---

e)

G.B. Hoghton, A.D. Keedwell

256

p = 1 mod4 and 2 is a quadratic non-residue, we have p = - 3 m o d 8 in the present case. So. -2 primitive implies 2 primitive and p = -3 mod 8 in the case rn > 1. Case 2. m = 1. In this case ! ( p - 1) is odd and p - 1 is even. Consequently, a must be even otherwise i ( p - 1)+ a would not be relatively prime to p - 1. Also. a must not have any of the q, as prime factors. When a is even and has none of the q, as prime factors, then 2 = g" has exponent $(p- 1) modp. Moreover, m = 1 implies p = 3 mod4 and a even implies that 2 = g" is a quadratic residue mod p. Consequently, we must have p = -1 mod 8. So, -2 primitive implies that 2 has the exponent f ( p - 1) modp and that p = - 1 mod8 in the case ni = I .

If p = 5 mod 8 then p = 1 mod4 and, for such primes, Friedlander has already proved that Dp is sequenceable. The primes p = 7 mod 8 relative to which 2 has the exponent :(p - 1) give a new class of sequenceable dihedral groups. In particular, of the twelve primes less than 200 which are congruent to 7 mod 8, the following nine are of the type required by Theorem 2: 7, 23, 47. 71, 79, 103, 167, 191, 199. (The primes 31, 127. 151 do not have -2 as a primitive root.) Table 1 Sequences satisfying Theorem 1, n m=4 ( 1 (2) 0 3 (3) 0 1 (4) 0 1 (5) 0 1 (6) 0 1 (7) 0 1 (8) 0 1 (9) 0 1 (10)'03 ( 1 1 ) " (1)

1 2 4 8 5 7 4 2 8 3 5 5 8 7 2 4 7 2 3 6 5 7 2 5 3 8 7 5 3 6 8 7 5 8 3 2 7 6 2 . 5 4 7 6 5 8 4 1 5 2 4 8 0 6 7 8 4 2 5

3 7 6 8 4 2 4 8 2 7 1

= 9,

6 6 3 3 6 3 6 3 3 6 3

' W e note that sequence (11) is the mirror image of sequence (10). It appears that when n = pq, p G q. sequences o f the form 0, p. . . . , . . . n - p always exist and i t is evident that each such sequence defines a mirror image sequence 0, n - p . . . , , . . . , p with differences p i + ~ - p iwhich are the negatives of those of its complementary sequence.

.

On the sequenceability of dihedral groups

257

Next, we consider the case when n is composite or is a prime (such as 31) for which the solution of the equation 2u=-1 m o d p is not a primitive root. A computer programme shows that, for all such values of n (n odd) up to n = 35 inclusive, the sufficient condition of Theorem 1 can always be met with rn = i ( n - 1) and that the number of suitable sequences 0, PI,Pz, . . . , Pn-l increases very rapidly indeed as n gets larger. Thus, for example, when n = 7 there is only one solution up to isomorphism (with rn = i(n - l)), when n = 9 there are eleven isomorphically distinct solutions and, for larger values of n, so many that the authors have not attempted to count them. This is very strong evidence for the conjecture that 0,is sequenceable for all odd values of n. The present authors have found the problem of trying to obtain a general method for the construction of sequences satisfying Theorem 1 for arbitrary choice of the odd integer n to be a surprisingly difficult one. To help others who may try to solve it, we list in Table 1 the eleven isomorphically distinct solutions for n = 9 with rn = 4 and in Table 2 we list one solution for each integer n = 3 mod 4 not covered by Theorem 2 and for each composite integer n = 1 m o d 4 (up to n = 35 inclusive). Table 2 Sequences satisfying Theorem 1 but not Theorem 2, n < 39, n odd ~

n = 9, m = 4. n = 11, m = 5 . n = 15, m = 7 . n = 19. m = 9 . n = 21,

m=10.

See Table 1 0 1 2 0 1 2 0 1 2 9 18 0 1 2

10 1 n = 2 5 . m=12. 0 7 n = 27, m=13. 0 1 25 n = 31. m = 1 5 . O 1 28 n = 33, m=16. 0 1 14 n = 35 m = 1 7 . O 1 23 21

Ashurstwood East Grinstead Sussex, England

17 2 20 2 20 2 21 2 25 2 30 31

4 4 4

3 3 3

4 13 4 11 4 14 4 11 4 17 4 20

3 18 3 22 3 24 3 23 3 27 3 33

6 1 0 7 9 6 8

7 9 6 11 5 12

6

8

5

6 18 6 16 6 16 6 20 6 25

8 21 8 23 8 24 8 32 8 34

9

5 8 5 14 10 13 8 12 7 15 13 10 14 17 11. 16 14 11 20

5 9 13 10 14 19 5 9 13 10 7 22 12 21 5 9 13 10 30 17 26 12 5 9 13 10 28 23 29 22 5 9 13 10 27 16 24 32

15

19 12

15 24

16 23

7

17 12

15 18 11 19 26 15 22 15 31 1.5 26

16

18 7 1 4 2 5 27 19 29 18 7 12 19 16 24 11 21 18 7 12 19 17 29 22 28

17 20 26 30 14 11

Department of Mathematics University of Surrey Guildford, England

G.B. Hoghton, A.D. Keedwell

'58

References [ I ] B.A. Anderson. Sequencings of certain dihedral groups, in: Proc. 6th S.E. Conf. o n Combinatorics. Graph Theory and Computing, Congressus Numerantium XIV (Utilitas Math., Winnipeg. 1975) pp. 65-76. (21 R. Friedlander, Sequences in non-abelian groups with distinct partial products, Aequationes Math. 14 (1976) 59-66. [3] B. Gordon. Sequences in groups with distinct partial products, Pacific J. Math. 11 (1y61) 1309-13 13.

141 A.D. Keedwell, O n the sequenceability of non-abelian groups of order

37 (1981) 203-216.

w.Discrete

Math.

[ 5 ] C. Vanden Eynden, Sequenceable countably infinite groups, Discrete Math. 23 (19778) 317-318.

Annals of Discrete Mathematics 15 (1982) 259-264 @ North-Holland Publishing Company

A COMBINATORIAL CONSTRUCTION OF THE SMALL MATHIEU DESIGNS AND GROUPS

D.R. HUGHES Dedicated to N.S. Mendelsohn on the occasion of his 65th birthday

Starting from the well-known, extremely simple (and folkloristic) designs 2 ’ 1 1 and 2’12 associated with the Hadamard matrix of order 12, a totally elementary construction technique leads to the Mathieu designs 4 1 1 and 4 1 2 and, more interestingly perhaps, to the Mathieu groups M I ] and M12,with such properties as their simplicity and their transitivity coming out as easy consequences of the combinatorics.

1. Introduction

We give here a particularly elementary construction of the small Mathieu designs, which has the interesting feature that it ‘automatically’ constructs the Mathieu groups for us as well, using only the most basic features of group theory. There are, of course, many ways known to construct these designs and groups, some of them folklore, and it is possible that this construction is not really new; but it seems never to have appeared in the literature. It is in fact possible to show the uniqueness of the designs by similarly elementary means, indeed extensions of the ideas used here. For details of this see [I].

2. The construction

We use more or less standard design theory notation. Thus we say that 9’ is a t-design for (u, k, A), or merely a t-(u, k, A), if it is a set of u points with certain distinguished subsets called blocks, such that each block contains exactly k points, and such that any set of t distinct points lies in exactly A common blocks; we also insist that no blocks be ‘repeated’. A structure is merely a set of points with distinguished subsets called blocks; it may have repeated blocks, and is uniform if every block has the same size, while it is a design if it is uniform and there are no repeated blocks. 259

260

D.R. Hughes

We let Zll be the well-known 2-design for (11,5,2); this design is unique (that is, with these parameters) up to isomorphism, and is variously known as a Hadamard 2-design, or more particularly, as a Paley design. It is also well known that there exists a 3-design Lf12for (12,6,2), again unique, which is an extension of Zll(that is, for any point P in Y12,the structure whose points are the points distinct from P and whose blocks are the blocks containing P, is in fact isomorphic to Yl1).In Y12the complement of any block is again a block. Then the following statements hold: (1) L I I= Aut(Zll) is 2-transitive (on points and also on blocks), is simple, and ] L I I=( 11 . 10 . 6 . (2) L I 2= Aut(TI2)is 3-transitive on points, is simple, and lLlzl= 12(Lll(. (3) Yll has a Singer group C,, with difference set D = {1,3,4,5,9}. (4) Zllhas a polarity cp. The proofs of these statements are either very well known or easy; in fact the same elementary techniques used in this paper will give them all, in little space. and (The crucial point is that Lfll is unique, which implies the uniqueness of Z12 the consequent transitivity properties of LIl and LIZ,from which the simplicity follows.) We also have the following statements: (5) The points of Zllare the elements 0, 1 , . . . , 10 of Cll and the blocks are the sets D + i (mod 11 of course). (6) The points of ZI2are the points of Yll plus a new point 00; the blocks of gI2 are: y U m, where y is a block of Zll,and y’, where y is a block of Yl1and y’ denotes the point complement of y in 91,. In general, we can write X, j , 2,. . . for the blocks of 2,,,and write 2’ for the block of Z12which is complementary to x. We use a well-known and elementary combinatorial result: (7) If Y is a t-(u, k, A), and s is an integer, 0 s < t, then any set of s distinct points of Y is contained in exactly A, blocks, where

==

A,

=A

(0 -

s)(u - s - 1).

(k - s)(k - s - 1)

* *

a

( u - t + 1) (k - t + 1)

(In fact this result is true in ‘uniform t-structures’, which is to say that repeated blocks can be allowed.) Now we construct a structure A,, as follows: (a) the points of A l l are the pairs (2, X’), where f is a block of Y I 2 ; (b) the blocks of A411 are the 2-sets [PIQ] where P and Q are distinct points of 9 1 2 ; (c) if (2, 2’) is a point and [P, 01 is a block, then ( f ,f’) is on [P, Q] if P and Q are both in X or both in f’. Then A l l has 11 points, since YIzhas 22 blocks in 11 ‘parallel’ classes; A l l

A combinaforial construction of Mathieu designs and groups

26 1

has 66 = (I;) blocks. Given a block [P, Q], the number of points (2, 2 ’ ) on [P, Q] is simply the number of blocks 2 of Lf12 on 2 distinct points, and that is five. So A l lis a uniform structure with u = 11 points, b = 66 blocks, and k = 5 points on a block. The automorphism group L I Zinduces an automorphisrn group of All, clearly, and we have the following theorem.

Theorem 1 . LIZinduces an automorphism group, which we call MI1, of which is 4-transitive on points.

~211,

Proof. First we note that LI2and MIIare abstractly isomorphic; for example LIZis simple and M I , is not the identity group. Let H be the subgroup of MII fixing (each of) 4 points (TI, jQ, (jjZ, jQ, (y3,j j ; ) , ( j j 4 , jji). Using (3) we choose these to be

If a is in H, then a fixes or interchanges the point sets jji and j j : , for each i. Hence if we take all possible intersections among the 8 sets above, we obtain sets of points which must be permuted by a. But we find that these intersections are all of size one, excepting exactly one which consists of 2 and 10. Hence a fixes both of 2 and 10, or interchanges them. In any case, blocks containing 2 and 10 must be fixed, and inspection shows that a must fix all the 8 blocks above. But then a fixes each individual intersection among the 8 sets, and so it fixes all points of Zl2,except perhaps 2 and 10. But it is trivial that then Q fixes 2 and 10 as well. So H = 1. Now the number of images of the ordered 4-set of points above, under M I 1 , is the index of H in MII, which is the order of MI1; this number is 12 * 11 . 10 * 6 = 11 10 . 9 * 8, which is also the number of ordered 4-sets of points in All.So Mll is transitive on ordered 4-sets, i.e., Ctransitive. Theorem 2. All is a 4-(11,5, 1). Proof. From Theorem 1, All has the property that any four points are on exactly A common blocks, and M l 1 is a uniform 4-structure (i.e., a Cdesign almost, but might yet have repeated blocks). Since b = A. = 66, we can compute A from (7) and find that it is 1, so there are not even repeated blocks. 0

262

D.R. Hughes

Theorem 3. A l l has an extension to A12,a 5-(12,6, 1). Proof. The extension is by adding a new point, ‘pasting’ it to all the old blocks,

and then adding the complements as blocks also. 0

It is in fact easy to show that in any S(12.6, l), the complement of a block must be a block, so At is the unique extension of All. Various proofs are known that any 4-(11,5, 1) is isomorphic to A l l (and in particular, see [l]), from which it would follow immediately that the automorphism group MI?=Aut(Alz) is 5-transitive on points. But we show this in a different way, which also reveals some of the group properties of Aut(Alz). is a substructure of A l l , and in such a way that Now we show that 2’11 Aut YI1= L I 1acts 2-transitively on t h e points of A l l (this in fact demonstrates that LI1is contained in MI,= LIZin 2 inequivalent ways). Let cp be a polarity of YI1 (see (4)). Each point of A l l is of the form (y U =, y’), where y is a block of 2’ll; we represent (y U m, y’) by the point Y = y‘ of Yll.The blocks of A l lare either of the form [ P , m], where P is a point of Yll, or of the form [P, Q], where P, Q are distinct points of YII;we represent [P, m] by the block p = Pv of YI1,and for [P, Q] we write p * 9, where p = P’, q = Q’. The incidence rules in this representation are the following: (a) Y on p e ( y Urn, y’) on [P,m ] a P on y ; since P = p v and y = Ya,this means Y on p in A l l e Y on p in Lfll. (b) Y on p * q e ( y U m , y ’ ) on [ P . Q ] @ P , Q are in y or P, 0 are in y ’ c j Y = y’ is in p = P* and q = Q’, or Y = y Q is in neither of p = P‘ and 9 = Q’. This means that p * q consists of those points which are either in both p and q or in neither p and q.

Theorem 4. LfI, is a substructure of A l l ,and the blocks of A l l are: (1) the blocks of YIl and ( 2 ) the sets p * q, where p, q are distinct blocks of Lfll, consisting of all poinfs in either both p and q, or in both p’ and 4’.Hence LL1is a 2-transitive subgroup of MII. Now the extension of A l l to is carried out by adding a new point (which we shall call r)and the complements of existing blocks. But clearly this will also extend the substructure Yll of A l l to a substructure (isomorphic to) Y12 contained in Alz.And AI2can then be viewed as constructed out of Y12 as follows: are blocks of Alz. (a) The blocks of (b) If (2,F’) and ( j j , 9’) are block pairs in Ylz, then (abusing language slightly), ( 2 n JJ)U (2’ n j i ’ ) and ( 2 n 1’)U(2‘ n j j ) are blocks of .Atlz. Hence LI2 is an automorphism group of M I 2 , so Aut Adlz is transitive. Thus Aut Alzis even 5-transitive.

A combinatonal construction of Maihieu designs and groups

263

To complete our analysis, we shall show that Aut All is exactly Mil; i.e., that every automorphism of All is induced by an automorphism of Y12. Since LIZis 4-transitive in its action on the points of All, it will suffice to show that the subgroup K of Aut Allwhich fixes 4 points is the identity. The blocks of All of t h e form y, where y is a block of Yll, will be called simple blocks, and the blocks y * z , where y and z are blocks of Y I lwill , be called double blocks.

Lemma 5. Let y be a simple block of All, and X , Y distinct points of y. Then there are precisely 2 blocks of All which meet y in exactly X and Y, and one of these blocks is simple, z say, and the other is the double block y * z. Proof. Certainly if y is a simple block and X,Y are points of y , then in Yll there is exactly one other block z on X , Y, so in Allthere is exactly one simple block z meeting y in X, Y only. But then the double block y * z also meets y in t h e points X and Y only. So it will suffice to prove that the number of blocks of All meeting y in exactly 2 points is 2. In .Mll we have h3 = 4 and h2 = 12. Let X, Y, Z be 3 points on y ; y is one of the 4 blocks on X , Y and 2 so there are precisely 3 blocks of Allwhich meet y in exactly the 3 points X, Y and Z. On X and Y there are 12 blocks; one of these is y , and for each choice of Z other than X and Y,2 on y, there are 3 blocks on X and Y which also contain 2.There are 3 choices of Z, so 9 blocks on X and Y meet y exactly one more time. Hence there are exactly 2 blocks of .MI, which meet y in the points X and Y only. 0 Now we consider the group K, and suppose a E K. The 4 points fixed by K can be assumed to be on a simple block y (since every 4 points of All are on one block, and MI, is transitive on blocks); then K must fix y and also the fifth point of y. Then a acts on the two blocks through exactly 2 points of y either by fixing both or interchanging them. If a fixes both members of each such pair, then a fixes all the blocks of Yll (for every block of Yll meets y twice), and hence, in LI1,a must be the identity, so a = 1 in Mll. On the other hand, a2fixes all the simple blocks, so by the same argument, a2= 1. Now let P be a point of All,P not on y , such that PZ P a ;a simple block z on P and Pa meets y in 2 points X and Y ; then z" contains P, Pa,X and Y, so za = z. Hence a fixes the fifth point Q on z , and Q is not on y. Now for any 3 points, X,Y,2 say, of y , there is a unique block of A l l on Q, X , Y, 2,and a must fix this block, so fixes the fifth point on the block. With X,Y fixed, the 3 choices of 2 gives us 3 such blocks XYZQ, hence 3 points besides Q, not on y which are fixed by a. Now a fixes at least 9 points of All, and it is easy to see that a = 1.

264

D.R. Hughes

Theorem 6. Aut A l l= Mil.

-

Corollary 7 . M12has order 12 . 11 10 9 * 8 , and MI2is simple.

Proof. Since Aut .Adl2= M12(by definition), and since a stabilizer of Aut is M ! , , of order 11 . 10 - 9 . 8 , the order of MI2is as given. To prove simplicity, suppose N t 1 is a normal subgroup of Mlz. If N2 MI],then N n MI,= 1 or Mil, since MII= L I Z is simple. If N n MI,= 1, then MI, acts as an automorphism group of N ; since N is transitive (it is a normal subgroup of a primitive group). IN] = 12. But the automorphism group of a group of order 12 cannot contain MI,.So N 2 MII,hence N = MI?. 0 We have demonstrated that L I 2= MIIoccurs as a stabilizer in M12,that is in a subgroup fixing one point and 4-transitive on the remaining 11. But, also, because of the ‘natural’ embedding of Lfl2 in AI2,LI2occurs as a subgroup of M12which is 3-transitive on the 12 points.

Corollary 8. M12contains MI1 in two inequivalent ways: as a stabilizer 4transitive on 11 points, and as a subgroup 3-transitive on 12 points. A number of other group-theoretic properties of M12can be deduced easily from .4t12:for instance. the 132 blocks of Jcc12 are naturally partitioned into 66 pairs (y, y ’ ) of complementary, o r ‘parallel’, blocks, and M12is transitive on these 66 pairs. Given a block y, any block z # y ’ , y meets y in 2, 3, or 4 points; the number meeting y 2 times is the same as the number meeting y 4 times, and this common number is 45. The number of z meeting y 3 times is 40. If ( z , z‘) has the property that z meets y 2 times, then z’ meets y 4 times, and it is easy to show that M l z ,acting on the block pairs, has rank 3: it is transitive, and the subgroup fixing a pair ( y , y ‘ ) has 2 additional orbits, one consisting of the 45 pairs (z, z‘) where z meets y 2 or 4 times, the other consisting of the 20 pairs ( z , z ’ ) where z meets y 3 times. Dept. of Mathematics Westfield College U.K. London “3.

Reference [I] T. Beth, Some remarks on D.R. Hughes’ construction of M12 and its associated designs. Finite Geometries and Designs, London Math. S o c . Lecture Note Series 49 (1980).

Annals of Discrete Mathematics 15 (1982) 265-272 @ North-Holland Publishing Company

EMBEDDINGS AND PRESCRIBED INTERSECTIONS OF TRANSITIVE TRIPLE SYSTEMS

C.C. LINDNER* and W.D. WALLIS** Dedicared to N.S. Mendelsohn on fhe occasion of his 65fh birthday Two problems concerning transitive triple systems (TITS) are discussed. It is shown that if y u = 0 or 1 (mod 3) and u 3 20 + 1, then there exists a 'ITS of order u containing a l T S of order u as a subsystem; and for every u 0 or 1 (mod 3) there exists a pair of lTSs of order u intersecting in exactly k triples, for any k in the range 0 s k s u ( u - 1)/3 except precisely

k

= U(U - 1)/3- 1.

1. Introduction

In what follows an ordered pair will always be an ordered pair (x, y ) where x f y . A transitive triple is a collection of three ordered pairs of the form {(a,b),

(a, c ) , (b, c ) } which we wiII always denote by (a, 6, c). A transitive friple system (TTS) is pair (S, T) where S is a set containing u elements and T is a collection of transitive triples of elements of S such that every ordered pair of distinct elements of S belongs to exactly one transitive triple of T. The number (St = u is called the order of the TTS (S, T) and it is well known that the spectrum for ITSs is the set of all u = O or 1 (mod3). It is a trivial exercise to see that if (S, T) is a TTS of order u then IT1 = u(u - 1)/3. Some examples of TTSs are the pairs (SI,T I),(S2,T2), and (S3, T3)defined as follows:

* Research supported by NSF Grant MCS 80-03053 and a grant from the Internal Research Assessment Committee, University of ,Newcastle. * * Research supported by an ARGC Grant. 265

266

C.C. Lindner, W.D. Wallis

The object of this paper is to give a complete solution to each of the following two problems: (1) given two numbers u and v each of which is 4 or 1 (mod 3) with u 3 2u + 1 , under what conditions does there exist a 'ITS of order u containing a subsystem of order u ?; and (2) for each u = 0 or 1 (mod 3). for which positive integers k is it possible to construct a pair of TTSs (S, Ti) and (S, T2) of order u such that IT, f l T21= k ? In particular, we show that (1) if u, u = 0 or 1 (mod 3) and u 2u + 1, then there exists a 'ITS of order u containing a TTS of order u as a subsystem; and (2) for every u = 0 or 1 (mod3), there exists a pair of TT'Ss of order u intersecting in exactly k transitive triples if and only if k E (0, 1,2, . . . , u ( u - 1)/3)\{u(u - 1)/3 - 1); i.e., if and only if 0 s k =su(u - 1)/3 excepf precisely when k = u ( u - 1)/3 - 1. It is worth remarking that both of the above problems have been solved for both Steiner triple systems and Mendelsohn triple systems. The reader is referred to [l, 3,4,5] for the appropriate solutions.

2. Embedding transitive triple systems

To begin with, so that there is no confusion, the 'fTS (S, T) is said to be embedded in the mS (Q, B) provided S C Q and T C B. Additionally, it is a trivial matter to see that a necessary condition for a ITS of order u to contain a subsystem of order u is u 2 2u + 1. We now prove that any 7TS of order u can always be embedded in a TTS of order u for every u 2 2u + 1 and u = 0 or 1 (mod3). Since 7Tss have the replacement property (i.e., if (S, T) is a 'ITS containing a subsystem (P, B) and (P, B*)is any TT'S, then (S, (T\B) U B*) is a TTS) it is only necessary t o construct for every u 3 2u + 1 and u = 0 or 1 (mod3) a TTS of order u containing a subsystem of order u. Lemma 2.1. A 7 T S of order u = 0 (mod 3), v f 6, can always be embedded in a 773 of order u = 0 (mod 3) for every u > 2u + 1. Since u = u = 0 (mod 3) and u > 2v + 1 both u/3 and u/3 are integers and u/322(u/3)+ 1. Hence by a result due to Hilton [2] there exists an idempotent quasigroup (Q, of order 4 3 containing a subquasigroup (P, of order v/3. Now set S = Q x {1,2,3} and define a collection of transitive triples

prod.

0)

0)

267

Prescribed intersections of transitive triple systems

T of S as follows: (1) ((x, l), (x, 2), (x, 3)) and ((x, 3), (x, 2), (x, 1)) belong to T for every x E Q; (2) if x f y , the six transitive triples ((x, l), ( x o y , 2), ( y , l)), ( ( y , l), ( y ox, 2), (x, I)), ((4 2), (x "Y, 3 1 7 (Y, 2))7 ((Y? (Y O X , 317 ( 4 2))7 ((43 1 9 ( X 0 Y 9 1x ( Y >3)), and ( ( y , 3), ( y o x , l), (x, 3)) belong to T. Clearly (S, T) is a 7TS of order 3(u/3)= u containing (as a consequence of the subquasigroup (P, of (Q, a subsystem of order 3(v/3) = v. 3

0)

7

0))

Lemma 2.2. A lTS of order v = 1 (mod 3), v f 7, can a l w a y s be embedded in a TTS of order u = 1 (mod 3) for every u > 2v + 1. Proof. Trivially ( v - 1)/3 and (u - 1)/3 are integers and (u - 1)/3 2 2[(v - 1)/3]+ 1. So, as in Lemma 2.1, let (Q, 0 ) be an idempotent quasigroup of order (u - 1)/3 containing a subquasigroup (P, of order (v - 1)/3. Let UJ be a U (Q X {1,2,3}). symbol which does not belong to Q x {1,2,3} and set S = {UJ} Define a collection T of transitive triples of S as follows: and ((x, 219 31, (1,1)) (1) (00, (x, I), (x, 211, ((4 m, ( 4 311, ((x, 3 1 9 ( 4 21, 9, belongs to T for every x E Q; (2) if x f y , the six transitive triples ((x, l), (xo y , 2), ( y , l)), ( ( y , I), ( y o x , 2), (x, I)), ( ( 4213 (x O Y9-319 (Y, 2119 ( ( Y , 21, (Y Ox, 319 (x, 2)), ((43), (x Y , 11, ( Y , 311, and ((y, 3), ( y o x , 1), (x, 3)) belong to T. As in Lemma 2.1 there is no problem in seeing that (S, T) is a 7TS of order 3((u - 1)/3)+ 1 = u. Since (P, is a subquasigroup of (Q, o), {m} U (P X {l, 2,3}) is a subsystem of (S, T) of order 3((v - 1)/3) + 1 = 0. 0)

(x7

O

0)

We now extend Lemma 2.1 to v = 6 and Lemma 2.2 to v = 7. The reason these cases cannot be handled by the constructions given in Lemmas 2.1 and 2.2 is because there does not exist an idempotent quasigroup of order 2. We can, however, slightly modify these constructions to extend these lemmas (which we now db). Lemma 2.3. A TTS of order 6 can always be embedded in a 7Ts of order u = 0 (mod 3) for every u > 2v + 1 and a TTS of order 7 can always be embedded in a 773 of order u = 1 (mod 3) for every u > 2v + 1.

Proof. If q 3 5 it is an easy exercise to construct a quasigroup (Q, of order q containing a subquasigroup (P,") of order 2 such that x o x = x for every x E Q\P. We now modify the constructions in Lemmas 2.1 and 2.2 in the following manner: In Lemma 2.1 define a 'ITS of order 6 on P x {1,2,3}, if x E Q\P define two transitive triples as in (l),and if x # y and borh x and y do not belong to P define six transitive triples as in (2). In Lemma 2.2 define a 0)

268

C.C. Lindner, W.D. Wallis

TTS of order 7 on {m} U (P x {1,2,3}),if x E Q\P define four transitive triples as in (1). and if x f y and both x and y do not belong t o P define six transitive triples as in (2). Our proof will be complete if we can handle the case where u - u f 0 (mod 3). The following lemma is the main ingredient necessary to take care of this case.

Lemma 2.4. Let v. u = 0 or 1 (mod 3), u 3 2v + 1. and u - v f 0 (mod 3). Then there exists a set A of size v and ( u - 2v - 1)/3 transitive triples B so that the set A U { b - a. c - a, b - c I (a, b, c ) E B } = Zu-u\{U} and such thaf ( u - v)/2 E A if id - v is even. Proof. In [3] it is shown that Zu-u\{O}can be partitioned into a set X of size v and ( u - 2v - 1)/3 3-element sets { a l ,bl, c , } , {az,bzrcz}, . . . , { a , b , CJ. t = ( u - 2v - l)/X such that a, + b, + c, = 0, i = 1,2, . . . , t, and ( u - v)/2 E A if (u - v ) is even. Set Y = {{al,br, cl}, {a2,b2,cz}, . . . ,{a,, b,, c,}} and partition Y in the following manner: Y1= {{al,b,, c,} I {-a,, -bl, -ci}E Y } and Y2= Y\YI. (We remark that Y , may well be empty.) Now form the graph (V, E) where V = Yz and E = {[x, y ] I there exists a pair (-a, a ) with a E x and -a E y } . Trivially (V, E) has degree at most three and so by Vizing’s Theorem [6] can be edge colored with at most four colors. Now removing the edges colored with two of the colors gives a spanning subgraph (V, 2’)of degree at most two. We now construct transitive triples from Y as follows: (1) For each pair {a, b, c}, {-a, -6, - c } E Y,form the transitive triples (0, a, a + b ) and (0, -a, -(a + b)). Now each triple in Yzhas degree 0. 1, or 2 in ( V , E’). We handle each case separately: (2) If {a, b, c} E Yz and has degree 0 in (V, E’), then at least one of a, b, c, say c, has the property that - c E X. In this case form the transitive triple (0, a, a + b ) or (0, b, a + 6). (3) If {a, 6, c}E Y2 and has degree 1 in (V, E‘), say x = { a , b, c } , y = {-a,u, v } and [x, y ] E E’, form the transitive triple (0, b, b + c ) or (0, c, b + c). And finally (4) if {a, 6, c } E Yz has degree 2 in (V, E’), say x = {a, b, c), y l = { - a , U I , VI>,YZ = 1-b, UZ, 4, and [x, YI], [x, y ~ El E‘, form the transitive triple (a, 0, a + c ) or (b, 0, b + c). We take B to be the collection of transitive triples in (l), (2), (3) and (4). Take A to the set X except in (2) when - c E X ; in this case replace - c with c. It is immediate that the sets A and B have the required properties, completing the proof. 0

Lemma 2.5. Let u, u = 0 or 1 (mod 3), u 2 20 + 1, and u - v f 0 (mod 3). m e n any T T S of order v can always be embedded in a 7 T S of order u. Proof. Construct sets A and B as in Lemma 2.4 and let (V,t ) be a TTS of order u where V n Zu-u= 8. Denote t h e elements in A by A = {x, xz, . . . , x u } .

Prescribed intersections of transitive triple systems

269

Let a be any 1-1 mapping from A onto V, set S = VUZ.-, and define a collection T of transitive triples of S as follows: (1) lG T ; (2) if (a, b, c) E B, t h e u - v transitive triples (a + i, b + i, c + i), i = 0,1,2,. . . , u - u - 1, belong to T; (3) if xj E A, the u - u transitive triples (i, xja, xi + i) E T where i = 0,1,2) . . . , u - u - 1 . It is straightforward to see that (S, T) is a 'ITS of order u and, of course, (S, T) contains as a subsystem the TTS (V, t) of order u. 0 Combining Lemmas 2.1, 2.2, 2.3 and 2.5 gives the following theorem.

Theorem 2.6. Any TTS of order u can always be embedded in a TTS of order u foreueryu2=2u+1 a n d u = O o r 1 (mod3).

3. Intersections of transitive triple systems

To begin with, since a ITS of order u consists of u ( u - 1)/3 transitive triples, if (S, T I ) and (S, T2) are a pair of TTSs of order u then ITl f l T21E {O, I , 2, . . . , u(u - 1)/3}. A bit of reflection shows that ITl fl T2(= u(v - 1)/3- 1 is impossible and so k E (0, 1,2, . . . , u = u(u - 1)/3}\{u - 1) is a necessary condition for two TTSs of order u to have k transitive triples in common. In this

section we will show that this obvious necessary condition is, in fact, sufficient. In what follows we will set I [ u ] = {0,1,2, . . . , u = u ( u - 1)/3}\{u- 1) and write J [ u ] = {k 1 there exists a pair of lTSs of order u with exactly k transitive triples in common}. Using this vernacular we will show that J [ u ] = I [ u ] for all u = 0 or 1 (mod 3). Since J [ u ] C I [ u ] , the proof entails showing that I[u] J[uJ. Our technique of proof is recursive and so, as is the case with most recursive constructions, we must begin by determining J [ u ] for some small values of u by ad hoc methods. In our case 'small' means u = 3, 4, 6, 7, 9, 10, 12, 13, 15, 16 and 18.

Lemma 3.1. J [ u ] = I [ u ] , u E {3,4,6).

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Lemma 3.2. J [ c ] = I [ c ] , u E (7.9).

Proof. Let (S. T) be a Steiner triple system and let us define t T = {(a.b, c ) I {a,b, c } E T and a < b < c ) , and - T = {(c.b. a ) I { a , b, c ) E T and a < b < c}. Now let (S, T I ) .(S, TJ. (S, 7'3) and (S, T,) be Steiner triple systems. Then each of (S. +TIU - T2)and (S, +T,U -T,) is a TTS and furthermore I(+T1U - T,) n (+ T3LJ- T,)( = 1 T , fl TI/+ 1 T2n TdI. As a consequence, since there exist pairs of Steiner triple systems of order 7 intersecting in 0. 1, 3 or 7 triples and pairs of Steiner triple systems of order 9 intersecting in 0. 1.2.3,4,6, or I2 triples [ 5 ] we have at once that Z[7]\{5,9, 11, 12) C J[7] and 1[9]\{11, 17,20.21.22} C J[9]. Now if (S, T) is any Steiner triple system and we define any 7Ts of order 3 on each triple of T the result is, of course, a ITS. Since {0,2) = 5[3]it follows at once that (0, 2, 4, 6, 8, 10, 12, 14) C 5[7] and (0, 2, 4, 6. 8. 10, 12. 14, 16, 18, 20, 22)C J[Y]. Hence 1[7]\{5,9, 1 l ) C J[7] and 1[9]\{11. 17, 19. 21)C 491. We now handle these exceptions. {5,9, 11) C J[7]. Let (S, A) and (S, B) be the following pair of Steiner triple systems: A = ((1. 2 , 3 ) , { I , 4, S), {1,6.7}, {2,4,6), {2,5,7), {3,4,7), {3,5,6}) and B={{1,2,4). {2,3,5), {3,4,6), {4,5.7}, {1,5,6}, {2,6,7), {1,3,7}}. Set T I = (+A) u (-A), 7 - 2 = ( [ ( + A )u ( - W \ W , 2,3), (1,4,5). (4,2, 1)))u {(2, 1,3), (4, 1,5), (1,452))- TJ= (+A) U (-B), T4 = (T3\{(1,2,3), (1.4,5), (4,2, I))) U ((2. 1,3), (4, 1. 5). (1,4.2)). and T5= (7',\{(1,2,3), (1,4,5), (1,6.7), (4,2, l), ( 7 , 6 2)))U ((2, 1-31? (4, 1,5), (1,7,6), (1,4,2), (6,7,2)). Then IT, n T2J= 5 , iT3n T4)= 11 and JT3nT51= 9. (11, 17, 19,21}C J[9]. To begin with, let (S, A) and (S, B) be a pair of Steiner triple systems of order 9 having exactly one triple in common (see [ 5 ] , for example). say {1,2,3}. Set T I =(+A)U(-A) and T,= ([(+A) U (-B)]\W, 2,3), (3,2, I))) U {(1,3,2), (2.3, 1)). Then IT, n T21= 11. Now let (S, T) be the 'ITS given by T = {(1,2,3), (4,5,6), (7.8, Y), (1,4,7). (2.4,s). (3,6, 9). (1,5. 9). (2,6,7), (3,4,8), (1.6,8), (2,4,9), (3,5,7), ( 6 2 . I). (9.5,J). (8,7.3). (7,4. I), (8,5,2), (9.6,3), ( 5 3 , l), (Y,7,2), (8,6,4), (9,8, I), (4,3, 2). (7,6.3}. Define T3 = ( n { ( l , 2,3), (5,3, l), (4,3,2)}) U ((3, 1,2), (5, 1,3). (4.2.311, TJ= (T3\{(2,5 , 8 ) , (8,5,2))) U {(2,8,5), (5,8.2)), and T5 = (TJ\{(~. 6,9), (9,6,3)})u ((3,546)- (6,9,3)). Then 1 T n T51= 17, IT n T41= 19. and 1Tf-I T31= 21. 0

Lemma 3.3. J [ u ] = Z[u], u E {lo, 12, 13, 15, 16, 18).

Prescribed intersections of transitive triple systems

27 1

Proof. It is a routine matter to construct PBDs of orders 10, 12, 13, 15, 16, 18 each of whose blocks has size 3, 4, or 6 and each of which has at least one block of size 4 or 6. If (P,B)is any such PBD of order u and we place a pair of ITSs on each block b E B intersecting in I ( b ) transitive triples, the result is a pair of TTSs of order u intersecting in C b E B I ( btransitive ) triples. Hence the intersection numbers which can be obtained from (P, B) are all C;=,x, + X$=ly, + zt where xi E J [ 3 ] ,y, E J[4], .zk E J [ 6 ] ,and B contains t blocks of size 3, f blocks of size 4, and s blocks of size 6. For each u E (10, 12, 13, 15, 16, 18}, regardless of the PBD used, a bit of routine computation shows that J[u]= I[u]. 0 Lemma 3.4. J[u]= I [ u ] ,for all u 2 9 .

Proof. We can always write u = 2u + 1 or u = 2u + 4 where u = 0 or 1 (mod 3). We handle each case separately. u = 2u + 1. Let (V, rl) and (V, f2) be any pair of TTSs of order u and (Y and p any two 1-1 mappings from Zu-u\{O} onto V. If (Y and p agree on x elements, then the construction in Lemma 2.5 gives a pair of 'ITSs (S, Tl) and (S,T2)such that IT, n Tzl= x ( u - u ) + Ifl f l r2(. Now if u 3 9 any k E I [ u ] can be written in the form k = x(u - u ) + y, where x E (0,1,2, . . . , u}\(u - 1) and y E J [ u ] . Since a and /3 can agree on x elements if and only if x E (0, 1,2,. . . , u}\(u - 1) we are through. u = 2u + 4 . Let a, b and c be distinct elements of Zu-u\{O} such that a + b + c = 0 and 2 a f 0, 2 6 2 0, 2cZ 0. Also let Bl = ((0, a, a + b)}, B2= ((0,b, a + b)}, and A = Zu-u\{O, a, b, - c } . Then each of the pairs A, B1and A, B2 have the properties in the statement of Lemma 2.4. Since the transitive triples (0, a, a + b) and (0, b, a + 6 ) are distinct, any two transitive triples of the form (i, a + i, a + b + i) and ( j , b + j , a + b + j ) are also distinct. Now let (V, t l ) and (V, t2) be any pair of TTSs of order u and (Y and p any two 1-1 mappings from A onto V agreeing on x elements. Now use the construction in Lemma 2.5 to construct (S, T I )using (V, tl), a,and A, B = B1 or Bz;and (S, T2)using (V, fz), p, and A, B = B1or B2.This gives a pair of 'ITss such that I Tl r l T2)= (x + y)(u - u ) + Itl f l f2(, where x E (0, 1,2, . . . , u}\(u - 1) and y = 0 if B is different in the construction of Tl and T2 and y = 1 if B is the same. As in the first part of this proof, since u 2 9 any k E I[u] can be written in the form k=(x+y)(u-u)+z, where x E ( 0 , 1 , 2 , . . . , u}\(u-l}, yE(O,l}, and z E J [ u ] . Since (Y and p can agree on x elements if and only if x E (0, 1,2, . . . , u}\{u - 1) and y can always be either 0 or 1 we are through. 0 Combining Lemmas 3.1, 3.2, 3.3 and 3.4 gives the following theorem.

Theorem 3.5. J[u]= I [ u ] for every u = 0 or 1 (mod 3). 0

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Remark. The authors are well aware that Lemma 3.3 can be expanded to J[u] = I [ u ] for all u 2 10. However, a general argument using PBDs is really no shorter than the argument in Lemma 3.4. The principal reason, however, for

Lemma 3.3 is the chance to illustrate an application of Theorem 2.6 (i.e., Theorem 2.6. other than being of interest in itself, has some no-nonsense applications). Department of Mathematics Auburn University Auburn, AL, USA Department of Mathematics University of Newcastle New castle, Australia

References (11 J. Doyen and R.M. Wilson, Embeddingsof Steiner triplesystems, Discrete Math. 5 (1973)22%239.

[ 2 ] A.J.W. Hilton, Embedding an incomplete diagonal latin square in a complete diagonal latin

square, J. Combin. Theory Ser. A 15 (1973) 121-128. [3] C.C. Lindner and D.G. Hoffman, Embeddings of Mendelsohn triple systems, Ars Combin. 11 (1981) 26-5-269. (41 C.C. Lindner and D.G. Hoffman, Mendelsohn triple systems having a prescribed number of triples in common, European 3 . Combin., t o appear. IS] C.C. Lindner and A. Rosa, Steiner triple systems havinga prescribed number of triples in common, Canad. J. Math. 27 (1975) 1166-1175; Corrigendum, ibid. 30 (1978) 896. [6] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz 3 (1964) 25-30.

Annals of Discrete Mathematics 15 (1982) 27S292 @ North-Holland Publishing Company

ON LINKED ARRAYS OF PAIRS Rudolf MATHON* Dedicated to N.S. Mendelsohn on the occasion of his 65th birthday

New combinatonal objects called linked arrays of pairs and linked triangular designs are studied in connection with association schemes and self-orthogonal latin squares. Constructions are given based on finite fields, direct products and pairwise balanced designs. A complete constructive enumeration and analysis is presented of such arrays with orders not exceeding 7.

1. Introduction

A t-(u, k, A ) design is a pair (V, g),where 9 3 is a collection of k-subsets, called blocks, from a u-set V such that each t-subset of V is contained in exactly A blocks of 9. We define a linked array of pairs (LAP) A to be a (s)x t array on the 2-subsets (blocks) from V satisfying the following conditions: (i) Every column of A forms a 2-(u, 2, 1) design. (ii) If R , ( x ) is the set of all rows with blocks containing an element x E V in the ith column of A, then the blocks in Ri(x) of any other column j form a 1-(u - 1,2,2) design Gii(x). The parameters u 2 4 and t 2 2 are called the order and deprh of A, respectively. G,(x) is called the derived graph of x in column i with respect to column 1-

A LAP(u, t ) A is said to be proper if there are no repeated blocks in any of its rows. If, in addition, the blocks of any row are mutually disjoint, then A is called pure. A LAP(u,2) can be conveniently expressed as an edge-labelling of the complete graph K, on V by interpreting the blocks of one column as edges and the blocks of t h e other column as labels. An edge ( i , j ) receives the label (k, 1 ) if and only if (i, j ) and ( k , I ) appear in the same row of A. So, for example, the unique LAP(5,2) (see Table 4 of Appendix A) yields an edge-labelling of Ks depicted in Fig. 1. * Research supported by NSERC Grant No. A86.51. 273

R. Mathon

274

Fig. 1.

Note, that as a consequence of (ii) the labels associated with the u - 1 edges incident with a given vertex x E V form a regular graph of order u - 1 and valency 2 which is a union of disjoint cycles. Let $23 denote a collection of ut ( u - 1)-subsets (blocks) from a set W, I W( = (2”). If the blocks of SB can be partitioned into t groups of u blocks each such that any two distinct blocks of the same group have one element in common, and any two blocks belonging to different groups have either 0 or 2 elements in common then (W, 58) is called a system of linked triangular designs (LTD) with parameters L’ and c.

Lemma 1.1. Every element of blocks in every group.

a system D of LTDs is incident with exactly two

Proof. Let ri be the number of blocks in a given group g which contain the element yi E W, 1 C i S w, where w = IWl = (;). Since any two blocks of g have one element in common, we have

C ri = u(u - I ) , W

i= 1

2 ri(ri- 1) = v ( u - 1). Y

i=l

Hence,

implying that r, = 2 for every i, 1 =si S w. An equally easy counting argument shows that any given block of one group is disjoint from exactly one block of any other group. A system D of LTDs is said to be in standard form with respect t o some u, the ith block of g and the ith block of any group g if for every i, 1 ==is

On linked arrays of pain

275

other group in D are disjoint. Without loss of generality we may assume that g is the first group in D. We note that the u blocks of every group in D form a partially balanced incomplete block design based on a strongly regular graph of triangular type (see D1). Proposition 1.2. LAP(u, t ) and LTD(v, t ) represent equivalent concepts. Proof. We will show how to construct from a LAP(v, t ) A with elements V = {xl, . . . , x,} a system D of LTD(u, t ) on W = { y l , . . . , yw}, w = Q ) , and vice versa. Label the rows of A by distinct elements from W, and label the blocks of every group in D by distinct elements from V. A + D: Block xj of the ith group in D is formed by the v - 1 row labels which correspond to Ri(xj)in A. D-+A: Block y j of the ith column in A contains the row labels attached to all blocks of the ith group in D which are incident with yi. By Lemma 1.1, y j contains exactly two elements.

The relationship between A and D reflects the duality which exists between the elements and blocks in a block design. From now on, depending on the application, we will use one or the other representation, keeping in mind their equivalence. Two LAPs A1(Vl) and Az(V2)of order v and depth t are isotopic if there exist a row permutation p E S@), a column permutation T E S,, and t bijections aj : Vl+ V 2 , j= 1 , . . . , t, such that for all i, j , 1S i S (;), 1 S j S t, thecell a: of A l contains the block (x, y ) if and only if the cell aE(i),r(j) of A2 contains the block (aj@), a j ( y ) ) .The ( t + 2)-tuple of maps (p, T, al,. . . , at)is called an isotopisrn of A 1 onto AZ. Two systems of LTDs D1(Wl,B1)and D2(W2,3,)are isomorphic if there exists a bijection a : Wl + Wz such that B E a1if and only if a ( B )E BZ.It is a consequence of the block intersection pattern that a maps groups of D1onto groups of D2. Moreover, two LAPs are isotopic if and only if the corresponding LTDs are isomorphic. Isotopisms (isomorphisms) of a LAP (LTD)onto itself are called autotopisms (automorphisms). They generate the autotopism (automorphism) group Aut. Let D be a system of LTD(v,2) in standard form on the element set w = { Y l , . . . ,yw}, w = (2”). Associate with D an undirected graph G ( D )on W with edges given as follows. Two vertices y , y j E W are joined by an edge if and only if there exist two blocks in D which intersect in {yi,yj}. Since isomorphic LTDs have isomorphic graphs we may use G ( D )to introduce some easily computable invariants for D. However, these invariants are not neces-

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sarily complete since isomorphism of the graphs does not always imply isomorphism of t h e underlying LTDs. From the definition of a system of LTDs we can easily deduce the following properties of its graph.

Proposition 1.3. If G ( D )is the graph of a system D of LTD(v,,2 ) then: ( i ) G ( D ) is regular of order ( 5 ) and valency 4. (ii) The elements of the xth block of the ith group in D induce in G ( D ) the derived graph Gii(x), j # i, 1 i, j 2. (iii) The derived graphs induced by the blocks of a group in D are mutually edge-disjoint and their union couers all edges of G ( D ) . (iv) The derived graphs induced by any two blocks from different groups in D have at most one edge in common. (v) The automorphism group of D is a subgroup of the automorphism group of G(D).

It is convenient to express the blocks of D as cycles of the derived graphs

G,(x) they induce in G ( D ) .For example, the cycle form and the graph of the

unique system LTD(S, 2) are shown in Figs. 2(a) and (b), respectively. LAPs and LTDs arise naturally in connection with many interesting combinatorial configurations such as triangle-free strongly regular graphs, biplanes, self-orthogonal latin squares, etc. Moreover, we believe, that they form a rich class of objects worth investigating in their own right. In the next section we study various properties of LAPs and derive upper bounds o n their depths. Section 3 concerns the relationship between LTDs and association schemes. In Section 4 we investigate problems related to the existence of LAPs. Constructions are given for pure LAP(u, fmax) of maximum depth whenever u is a prime power and lower bounds on t,,, are discussed for 1

0 (1, 2 . 4 , 3 ) 1 (1,7,5,6) 2 (2.8.9,s) 3 ( 3 . 8 , 6, 10) -I (4.7, 10. Y)

5

(S,6, 10.9)

(3.8,9,4) (1. 7, 10.3) (2.4,7,5) (I.?,& 6)

3

(a) Fig. 2.

7

277

On linked arrays of pairs

the general case. Section 5 contains a constructive enumeration and analysis of LAPS for orders u, 4 d u =s7. In Section 6 we list some interesting open problems. Finally, Appendix A summarizes computational results.

2. Properties and bounds

A LAP(u, 2) A can be viewed as an arrangement of all ordered pairs from V into a u x u square array L, = (Iij) such that (i) 1" = (i, i), (ii) if lv = (k,I), then Iji = (I, k), = (i, j ) and uR = (k,I) for some row r of A. (iii) lij = (k,I) or (1, k) iff There is a great deal of freedom in assigning directions to the pairs in Lv. Since every row and column of L, contains every element of V exactly twice it might be possible to order the pairs in such a way that every element appears exactly once in the ith position of any row and column of L, i = 1,2. Then L, is equivalent to a latin square which is orthogonal to its transpose. To illustrate this process we represent in Fig. 3 the LAP(5,2) as a square in two different ways. It is easily verified that the square in Fig. 3(b) corresponds to a self-orthogonal latin square. When is a LAP(u, 2) A equivalent to a self-orthogonal latin square ? Is there an efficient algorithm for orienting L, from a given A ? We begin with a characterization of a self-orthogonal L, in terms of the corresponding LTD(v, 2) D and its graph G(D). From Proposition 1.3 it follows that every edge of G ( D ) appears exactly once in the derived graphs Gij(x)induced by blocks of a group i in D. Therefore, orienting the edges of G ( D )induces edge-directions in the derived graphs Gij(x),i # j . 0

1

2

3

4

0

00 42 43 21 31

24 11 30 40 32

34 03 22 41 10

12 04 14 33 20

13 23 10 02 44

00 42 34 21 13

(a) Fig. 3.

1

2

3

4

24 43 12 31 11 30 04 23 03 22 41 10 40 14 33 02 32 01 20 44

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278

The following result is a direct consequence of the duality between LAPs and LTDs. Proposition 2.1. L, is self-orthogonal if and only if there exists an orientation of G ( D ) such that all Gji(x)are unions of directed cycles. Testing whether or not G(D) is orientable requires time O(u2).This can be done as follows. We start by orienting a cycle in Glz(x)which induces directions on the corresponding edges in the derived graphs of the other group. Continuing to orient further cycles and alternating between the groups, we stop if either a cycle gets opposing directions in which case G ( D ) and L, are not orientable, or all derived graphs become unions of directed cycles, implying that G ( D )and L, are orientable. For every connected component of G ( D )we may orient one cycle arbitrarily. Hence an orientable G ( D )with r connected components yields 2' distinct self-orthogonal L,. The algorithm can be translated to work directly on L , by exploiting the correspondence between the derived graphs Gji(x)and the rows and columns of L,. To do this we label the derived graphs of each group in D by the numbers V = (0, 1,. . . , u - 1). Let y be the vertex common t o G12(i)and GI2(j).Denote by ( x , y ) and ( y , z ) the directed edges to and from y in GI2(i). respectively. If C 2 , ( k )is the derived graph containing (x, y ) , and if Gzl(f) contains (y. z), then 1, = (k,I ) . The LTD(S, 2) in Fig. 2(a) can be oriented by directing all its cycles from left to right. The corresponding square is given in Fig. 3(b). Self-orthogonal latin squares have been studied by Mendelsohn [ 101 and others [2, 5 , 71. In Section 4 we will use them to help settling the existence problem for LAPs of depth 2. We require the notion of a sub-LAP. Let A be a LAP(u, t) o n the set V. A (;) x t subarray A' of A is called a sub-LAP(u, f ) if every column of A' forms a 2-(0,2, 1) design with elements from a u-subset U C V. A sub-LAP A' of A is called aligned if for any two distinct columns i. j and any x E U there exists a y E U such that the derived graphs Gjj(x)have the vertex sets v \ { y } and U\{y} in A and A', respectively. Proposition 2.2. Let A be an LAP(u, t ) confaining a sub-LAP(u,t ) A'. u < u. Then va(2t-l)u-t+1.

(3)

If A' is aligned, then u a (2t - 1)u + 1

(4)

On linked arrays of pairs

279

Proof. Represent A as a system D of LTDs on the set W, 1 Wl = (5). The rows of A’ in A correspond to a subset W’ C W, 1 W l = (f).Denote by yi the union of elements from W\W’ which appear in the u blocks containing u - 1 elements from W’ in the ith group of D.Then, since I Y (= u(v - u ) , and I Y n Y , I s 2 u for l S i < j G t , we have

(;)-

(;)a

tu(v- u)-2u

If A’ is aligned, then I Y , r l

(;) - (;)

(3

= fu(v- u - t + 1).

ql=0 for any i f j , and so

3 ru(v -

u).

It is easily verified that (5) implies (3) and that (6) implies (4).

If equality holds in (3) or (4), then the corresponding sub-LAP is called maximal. The only known examples of LAPs with maximal sub-LAPS are derived from self-orthogonal latin squares. In [5] a self-orthogonal latin square of order v = 3u + 1 with a self-orthogonal subsquare of order u is constructed for every u # 2,3,6. This implies the existence of a LAP(3u+ 1,2) with a maximal aligned sub-LAP(u, 2) for every u 3 4, u # 6. In the remainder of this section we shall estimate the maximum depth t of a LAP(u, t ) . We begin with pure LAPs. Proposition 2.3. In a pure LAP(v,

t)

of order v 3 4,f

G

[v/2J

Proof. The inequality follows from the fact that in a pure LAP the 2-subsets of any row are mutually disjoint.

In Section 4 we will construct pure LAPs of maximum depth every prime-power order u 2 4. The case of general LAPs is treated next.

f =

[v/2] for

Theorem 2.4. The depth t of a LAP(v, t) satisfies the following inequalities (i) I f v = 8 k - 2 1 , O S l G 3 , k 2 1 a n d v 2 6 , t h e n t s k(8k - 41 + 3) + 283,

where 831 = 1 if 1 = 3 and 831= 0 otherwise.

(7)

280

R. Mathon

(ii) If u

= 4k

+ 21 + 1, 0 s 1 =s1, k 2 1

and u k 7 , then

Proof. Let 93 be a collection of k-subsets from a w-set W such that any two distinct subsets intersect in at most A elements. It has been shown by Schonheim [ 1 1 1 that

We will apply this bound to t h e corresponding dual LTD(u, t) D. The blocks of D are ( u - 1)-subsets from a ($')-set which pairwise intersect in at most 2 elements. Consequently,

The final inequalities (7)-(9) result from a careful but straightforward examination of ( I 1). Table 1 lists values of TI and T2 bracketing the maximum depth t = t,,, of a LAP(u, r ) , T ,c t,, s Tz, for orders u. 4 =s IJ s 13. Here TI is found from the constructions i n Sections 4 and 5 , T2is found by evaluating the right-hand sides of ( 7 x 9 ) . The starred entries correspond to values of l,,,=. We note, that since (10) does not take into account the actual structure of LTD(v, t) the quadratic asymptotic behavior of T2 as a function of u is probably incorrect. We conjecture that t,,, is bounded both from above and from below by a linear function of u. Table I L!

4

5

6

7

8

9

10

11

12

13

3. Association schemes A finite set X o f n elements together with rn + 1 symmetric relations Ro= I, R , . . . . , R , defined on X is called an m-class association scheme [l] if

On linked arrays of pairs

281

(i) For every x, y E X , (x, y) E Rifor exactly one i. ( z ,y ) E Rj (ii) If (x, y ) E Rk,then the number of z E X such that (x, z ) E Ri, is a constant pi independent of the particular choice of x and y . Two points x, y E X are called ith associates if (x, y ) E Ri.It is helpful to view an m-class association scheme as an edge coloring of the complete graph K,, where (x, y) is assigned color i if x and y are ith associates. Then (ii) says that edges of color i form a regular graph, and that the number of triangles of given type on a given base depends only on the colors of the edges. Let Ai be the adjacency matrix of the graph Giof ith associates (color). Then

Denote by Aj(i) the ith eigenvalue of Aj and let pi be its multiplicity, 1 c i, j S m. It is a consequence of (12) that Aj(i) and p ican be expressed in terms of the intersection numbers p;. A system D of LTD(u, t ) may give rise to a 3-class association scheme d on its blocks with relations being induced by block intersections. Define two blocks of D to be ith associates in d if they have i - 1 elements in common, 1C i C 3. The intersection numbers of d can be expressed in terms of u, t, and two other parameters a and P : nl = t - 1 ,

n2 = u - 1 ,

n3 = (t - l)(v - 1) ,

Pi1 =

p:2 = 0

pi3= t - f f - 2 , p$=(t-2)(u-2)+a,

a,

p&=O,

0 p & =u - 2 , p:1=

p:1=

9

P,

Pi2 = 0 ,

, pL= u - I , p:2

=0,

p:3=

t - 1,

p:=o,

p:3 = ( t - l)(u - 2),

1, p % =u - 2 ,

p:3= t - P - 2 ,

p:2=

p:3=

(t-2)(u-2)+/3.

Here ni = pz is the valency of Gi. In order to see which values of u, t, a and P are admissible we will apply the known necessary conditions for the existence of 3-class association schemes [3, 81.

Theorem 3.1. Let d be a 3-class association scheme with parameters (13). Then there exist non-negative integers k, 1 and P such that (i) klP is even, (ii) cy = P + k - 12 0, Pv = (I - l)(k + l), t = k l + 1,

282

R. Mathon

(iii) Ir(u - l ) / ( k + I ) is a positive integer, (iv) k Z- 1 3 0.

Proof. (i) The order and valency of Gicannot be both odd. (ii) follows from the relations ng:,= njpsl.(iii) expresses t h e fact that the pi’s must be positive integers. (iv) is implied by the so-called Krein conditions (see [8] for details). The eigenmatrix P and multiplicities pi of d can now be expressed with help of k, I and /3: rl

kl

u - 1 k l ( u - 1)1

We will exhibit two sets of parameter combinations for schemes which are known to exist in systems of LTDs. (1) If I = O , then p = O , t = k + l , a = k - 1 and p l = u - l , p 2 = k r p 3 = k ( u - 1). The association scheme is of a product type K , @ K,and is induced by the block intersections of any pure LTD(u, t). All values k , 1 S k S [u/2J - 1 , are admissible by Proposition 2.3. (2) If I = k + l and p = l , then t = k Z + k + l , a = O and u = t - l . Since 2k + 1 divides k ( k Z + k + l ) ( k * + k ) only if k = O , 1 , 2 and 7, we see that u = 0 , 2 , 6 and 56. Except for the trivial case u = 0, these schemes are related to Moore graphs of diameter two (see [6, 81). The scheme is induced by the second associates of the Petersen graph for u = 2, and of the HoffmanSingleton graph for u = 6. No scheme is known for u = 56. The case u = 2 is too small to be related to LTDs; the association scheme for u = 6 is induced by the blocks of the unique LTD(6,7) (see Tables 4 and 5 of Appendix A for t h e listing and properties of the corresponding LAP(6,7)). We remark that there are more parameter sets which satisfy the conditions of Theorem 3.1. So, for example, if 1 = k is even and p = 1, then t = k Z + 1, a = 1, u = kZ- 1 and p l = p 3= ( k 2 + l ) ( k z - 2)/2, p2 = k. For k = 2 a scheme is known to exist (line graph of the Petersen graph) but is too small to be related to LTDs. For k 2 3 no examples are known.

4. Existence

t

We begin this section by constructing a pure LAP(u, I) of maximum depth for every prime-power order u 2 4.

= Lu/2J

On linked arrays of pairs

283

Lemma 4.1. Let u = 2t + 1 be a power of an odd prime, u > 3. If a is a primitive element of the Galois jield GF(u),then the (2) X t array

[{a'+i+p"a''i+p}, i = O , 1, . . . , t - I], 1 = 0 , 1 , ..., t - I , P = 0, (YO,. . . , (Ya-1

(16)

is a pure LAP(v, t).

bf. In GF(u) of odd order u = 2t+ 1, a' = -ao = -1. For p = we obtain the pairs (0, ?2a'"-' } in the ith column and the pairs ?a'+i-l{aj-i + 1, + I} in any column j # i. Here 2a = a + a and a{& y } = {a&ay}.Since 1 ranges over the set {0,1,. . . , t - 1) we see that 0 appears with every other element of GF(u) in the ith column, and Ri(0) of column j contain every nonzero element of GF(u) exactly twice. To complete the proof we observe that the t pairs of every row are mutually disjoint.

Lemma 4.2. Let u = 2", t = 2"-' for some n > 1. If (Y is a primitive element of G F ( u ) and Y = {PO,P1,.. . , is the subgroup of index 2 of the additive group of GF(u), then the (z)X t array [ff'{Pi+Pk,Pi+Pk+1},i=O,l,

. . . ,t-11,

I = o , I ,..., 2t-2, k = 0 , 1 , . ..,t - 1

(17)

is a pure LAP(u, t ) .

Proof. In GF(2"), 0 E Y, 1!Z Y. Hence Y + 1 is a coset of Y. Let L, = {I: a'(P + y ) = 1, P E Y},y = 0 , l . Then it is easy to see that L,,f l L1= 0 and U L1= {0,1, . . . , 2 t - 2). To show that (17) is a LAP it suffices to look at blocks in Ri(0)and R i ( l )of two distinct columns i# j . For pk = Pi-1 we obtain + + 1) in R,(O) of column i and j , the pairs (0, a'} and a'{&] +

respectively. Similarly, for P k # 0, 1 E such that ( Y ' @ ~+- ~P k )= 1, and for P k E Y,1 E L1such that a'@i-' + P k + 1) = 1, we obtain the pairs (1, a' + 1) and {a'(pi-l+&-I)+ 1, a'@;-1+ Pi-1 + 1) + 1) in R i ( l ) of column i and j , respectively. Since I ranges over the set {0,1, . . . , 2 t - 2) the axioms of a LAP are satisfied. As before, the t pairs of every row are mutually disjoint. Combining Lemmas 4.1 and 4.2 we obtain the following result. Theorem 4.3. A pure LAP(u, t ) exists for every prime-power order u 2 4 and 2 c t c 1421. We require a few more definitions. A k X u2 matrix S with entries aijE (1, . . . , u } is called an orthogonal anay (OA) of k constraints, u levels (strength

284

R. Mathon

2 and index 1) if every ordered pair occurs exactly once as a column in any 2 x v z submatrix of S. An OA(v, k) is equivalent to k - 2 mutually orthogonal latin squares of order u (see [4, p. 1901). An extension of a LAP(u, t ) A is a u2 x t array of pairs containing two copies of A and u constant rows of (x, x)’s, one for every element x. A is said to be orientable if the pairs of its extension can be directed so that A is the transpose of an OA(u,2t). An orientable LAP(v, t ) corresponds to a set of t - 1 self-orthogonal latin squares (see Proposition 2.1 for the case t = 2). The classical LAPs (16) and (17) exhibit a number of symmetry and regularity properties inherited from the underlying finite field. These properties are summarized in the following proposition which can be verified along the same lines as Lemmas 4.1 and 4.2. Proposition 4.4. Let A be the LAP(u, t ) on V = G F ( u ) given by (16) or (17). 7hen the following properties hold: (i) Aut A acts transitively on the rows and columns of A and is doubly transitiue on its elements. (ii) For any given i, 1 i =S t, and any divisor k 3 3 of u - 1 there exists a j such that Gij(x)is a union of (u - l)/k disjoint k-cycles for every x E V. (iii) A is orientable with all its pairs correctly directed.

We will make use of various results known for self-orthogonal latin squares to construct new LAPs. It is known [2] that self-orthogonal latin squares exist for all orders u f 2 , 3 , 6 . For u = 6 there exist two non-isotopic LAPs (see Tables 4 and 5).

Theorem 4.5. A LAP(u, 2) exists for every v 3 4. A large number of constructions for orthogonal arrays is based on direct products and singular direct products (see [4, 71). Both products are known to preserve self-orthogonality [7]. Consequently, the following two results hold. Theorem 4.6. If there exist orientable LAPS of order ul, u2 and depth t, then an orientable LAP( u1u2, t ) can be constructed.

Theorem 4.7. Suppose there exists an orientable LAP(uI+ u2, t ) containing an aligned orientable sub-LAP(ul,I ) . If there exist orientable LAPs of order v2, u3 and depth t, then an onentable LAP(vl+ u2u3. I ) can be constructed. Finally, we will look at recursive constructions based on pairwise balanced designs. A painvise balanced design (PBD) denoted by PBD(u; k l , . . . , km) is a

On linked arrays of pairs

285

collection 9 of blocks from a u-set V such that every pair of distinct elements from V is contained in precisely one block of 9 and every block of 9l has cardinality ki for some i, 1S i G m. Theorem 4.8. If there exist a PBD(u; k,, . . . , k,,,), and a pure LAP(ki, 1 ) for every i, 1 s i c m, then a pure LAP(u, t ) can be constructed. Moreover, if every LAP(ki,t ) is orientable, then so is LAP(u, t). Proof. Let V be the elements and B1,B2,..., Bb the blocks of PBD(0; kl,. . . , k,,,). Denote by AJ a pure LAP(ki,t ) on the ki = IBJ(elements of BI. Form a new array A by concatenating the A[, 1 = 1,. . . , b. Since in a PBD

it follows, that every column of A contains every pair from V exactly once. Given any element x E V let L(x)= {I: x E BI}.The LAPs AJ, 1 E L(x) induce a partitioning in A of the rows Rj(x)into IL(x)I sets Rf(x),1=si S f. Consider a column j of A, j # i. Since every AJ is pure the derived graphs G!y(x)are mutually vertex-disjoint. Consequently, A is a pure LAP(u, r ) in which the A, form a system of aligned pure sub-LAPs. From the construction it is obvious that if every A, is orientable, then so is A. The last theorem shows that LAPs are PBD-closed [12]. By combining the Theorems 4.3, 4.6,4.7 and 4.8 together with existence results for PBDs it may be verified that for any t there is a constant uI such that there exists a LAP(u, t ) for all u > uf. Upper bounds for the numbers uf can be obtained similarly as those for OAs [13].

5. Enumeration

Constructions based on products and PBDs are useful for obtaining asymptotic lower bounds on the number of non-isotopic LAPs. One method relies on the fact that LAPS resulting from such constructions contain many sub-LAPS which can be independently permuted. We will illustrate this approach on the PBD construction of Theorem 4.8. A subset Bf of blocks in a PBD forms a so-called fixing set if the only isomorphism fixing each block of 9, is the identity.

2%

R. Malhon

propodtion 5.1. Let afC 9 be a f i i n g set of blocks in a P B D ( v ; k l , . . . , k,,,). Suppose that for euery BI E 9, 1B,/ = k,, 1481 = b there exists a pure autoropismfree LAP(k,, t ) , and that the LAPS corresponding to Bf U {BI} are painvise non-isotopic for every BI E 9\58,. Then there exist at least

painvise non-isotopic pure autotopism -free LAP(v, t ) .

Let A, be the LAP(k,, t ) constructed on the elements of BI E 93, 1B,l = ki,and let A be the final LAP(v, I ) . Let ?/(A)denote a permutation of columns of A and let al(A)= (all,. . . , al,)(A)be a t-tuple of maps permuting the elements of A, in A . Since A is a concatenation of the A/ we have ?/?k = ?k?/, a p k = aka/,whenever I # k, and 7pk = akr, for any 1 =sI, k S b. Moreover, A' = cp(A)is a LAP(u, t ) for any composite map cp = lI?=' rpI.Since every block of the fixing set af is uniquely labelled by a pure autotopism-free A/, both A and A' are also pure and autotopism-free. Thus, an isotopism between A and A' preserves the constituent sub-LAPS A one by one. Consequently, A' is isotopic to A if and only if 71 = r2= . * * = 7 6 and a 1= a2 = . . * = f f b = E , where E is the identity. The number of all maps cp which are not of this form is given by (19). prod.

As an illustration of t h e above procedure we consider a projective plane of order 7 . It is a PBD(57; 8) with blocks ( 0 , 3 , 5 , 13,14,20,32,36)(mod 57). It has a block-fixing set of size 4 (for example, { 1 , 2 , 3 , 4 } is a dual element-fixing set). In the following corollary AF stands for autotopism-free.

Corollary 5.2. The existence of 5 painvise non-isotopic AF L A P ( 8 , 2 ) implies the existence of 2%(8!)'14A 7.7 x 10"' painvise non-isotopic AF LAP(57,2). Similarly. 5 pure A F L A P ( 8 , 3 ) yield 6%(8!)"'= 1.32 x loa' pure A F LAP(57,3). The technique of Proposition 5.1 can be generalized to include sub-LAPS which are not necessarily autotopism-free. By using this more general approach on the number recursively one may derive the asymptotic lower bound 2cu2r'og' of non-isotopic LAP(u. t ) . In the remainder of this section we present a complete enumeration of LAP(v, t ) for small orders u, 4 u 7 . The numbers N(u, t ) of non-isotopic LAP(/\,t ) are listed in Table 2. Moreover, N ( 6 , r ) = 1 for t = 4, 5, 6 and 7.

On linked arrays of pairs Table 2 V

t

2 3

4

5

6

7

1 0

1 0

2 4

6 2

A transversal of a system D of LTD(v, t) on the elements W is a ( v - 1) subset of W which intersects every block of D in 0 or 2 elements. The generation technique consists in a computer aided breadth-first search employing transversals in LTDs. We will now describe this search in more detail. Starting with f = 1, we generate all non-isomorphic LTD(u, f + 1) which contain a given system 0, of LTD(u, f ) according to the following algorithm (see [9]): (i) Find all transversals Bf of Df and partition them into orbits under the action of Aut Of. (ii) Employing the orbits in 8, find the collection 9, of all u-sets of transversals pairwise intersecting in one element. Partition 9, into orbits under Aut 0,. The orbit representatives are group candidates for the LTDs D,+lof the next level. (iii) Reject those Df+l which are isomorphic to LTDs generated before. For isomorphism testing use invariants such as the order of Aut and its orbits, the number and distribution of cycles in G(D,+l),orientability, etc. We make a few observations. From the way 9,and 9,are defined it follows that gfC 9f-l C * * . C B1.Moreover, by Lagrange's theorem we have

IAutf Dfcll

= lorbit

Aut 0, of grc!up of t!locks in 9,l

In order to appreciate the where AutlDf+lis the stabilizer of 0, in Aut complexity of the above search we list the sizes of orbits in which CB1 and 9, are partitioned under the action of Aut Dl = S, in Table 3. For orders u 2 8 a breadth-first search at the lowest level becomes infeasible Table 3 V

4

5

6

7

APBi AP91

4

15 6

72 12,180

70,420 12@,240,8402,1686

1

zxx

R. Mathon

because of the large sizes of and 9,. A more promising approach consists in a systematic backtrack search for Dz with isomorph rejection, followed by a breadth-first search for r 3 3. For order u 2 10 an exhaustive enumeration becomes prohibitive due to the combinatorial explosion.

6 . Open problems

We recall here some open problems suggested in the text. (1) Find a LAP which is not isotopic to any proper LAP. (2) Find a LAP containing a (maximum) non-aligned sub-LAP. (3) Improve the bounds on the maximum depth of LAP(u, t). (4) Construct more LTDs which are related to association schemes. In particular decide the existence of LTD(56,57). (5) Estimate the numbers uI which imply the existence of a LAP(u, t) for every t' > u,. (6) Prove the asymptotic lower bound 2 C U * ' on 1 ~ the r number of non-isotopic LAP(u, t ) . (7) Enumerate the LAPS of order 8 and find examples of LAP(l0,S) and LAP( 13.6).

Appendix A In Tables 4 and 5 of Appendix A we list all LAP(u, t), 4 s u s 7, together with their properties. Table 6 lists the known LAP(u, t ) with 8 S u =s9, and t = 4. The table headings are explained below: - D, = set of columns from Table 4 which generate the LAP(u, t ) . - (Aut,l = order of autotopism group of D,. - AP, = sizes of row-orbits under Aut,; s' denotes i s-orbits. - IAut,l = order of Aut, restricted to the columns of 9. - AP, = sizes of column-orbits under Aut,. - Ck = number of k-cycles in G(D,). - ID, = list of D2 generated by columns 12, 13, 23, respectively. -C = cycle sizes in the 0-block intersection graph of the dual LTD; k' denotes i &-cycles. -T = type of 4; p = pure, r = proper. - NO = number of non-equivalent orientations of 4 .If Dl is orientable its constituent columns have the correct directions in Table 4.

289

O n linked arrays of pairs

Table 4 The basic LAPS of order

0,

4s v 67

v=4

1 2 3 4 5 6

0=7

0

1

0

01 02 03 12 13 23

23 31 12 03 20 01

01 46 02 65 03 54 04 31 05 23 06 12 12 53 8 13 26 9 14 05 10 15 42 11 16 30 12 23 41 13 24 63 14 25 10 15 26 04 16 34 20 17 35 06 18 36 51 19 45 61 2 0 4 6 2 5 21 56 43 1 2 3 4 5 6 7

v=5

1 2 3 4 5

6 7 8 9

10

0

1

01 02 03 04 12 13 14 23 24 34

24 43 12 31 30 04 23 41 10 02

1

2

3

4

2 5 4 6 34 65 16 54 62 13 41 21 53 32 6 0 3 0 04 26 23 05 36 42 45 53 50 15 15 61 6 4 3 4 31 40 56 20 12 06 24 14 03 63 25 10 02 01

65 34 12 26 41 53 05

5

6

7

56 46 36 36 45 12 13 15 12 34 24 25 04 05 4 0 4 6 5 6 63 25 03 32 03 24 24 23 23 06 15 45 15 16 16 64 34 13 31 05 04 52 02 02 16 26 06 45 01 14 03 06 26 10 35 35 02 14 01

46 26 56 56 24 45 13 12 14 13 35 23 03 03 46 06 05 35 23 24 45 25 15 15 16 16 0 4 0 4 3 4 3 4 25 02 06 26 01 14 3 6 3 6 02 05 12 01

v=6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

8

0

1

2

3

4

5

6

7

8

9

A

01 02 03 04 05 12 13 14 15 23 24 25 34 35 45

23 14 15 25 34 05 04 35 24 45 03 13 12 02 01

24 15 45 13 23 03 25 05 34 14 35 04 02 01 12

25 13 24 35 14 04 45 23 03 05 15 34 01 12 02

34 35 12 15 24 45 05 02 23 04 13 01 25 14 03

35 45 14 23 12 34 02 25 04 15 01 03 05 24 13

45 34 25 12 13 35 24 03 02 01 05 14 15 04 23

24 13 25 15 34 05 45 23 03 04 35 14 01 12 02

35 45 14 23 12 34 02 05 24 15 01 03 25 04 13

34 45 15 12 23 35 02 25 04 14

45 35 24 13 12 34 25 02 03 01 15 04 05 14 23

03 01 05 24 13

R. Marhon

Table 5 TheLAPsoforder u . J S u ~ 7 u

No.

4

IAutd

4

1

1

01 01 01 07 01 03 05 06 07 08

48

5 6

1

7

2 i 2 3 4 5 6

AP,

C 4

3

6 10 15 1,2.4,8 21 3, 8 21 3.6" 37 37

360 72 24 24 126 18

15 3,12 3.4,8 3,4,8 21 3,18

40 120 8 42 6 42 6 3

2

01A 012 078 079 012 034

4 5 6 7

0123 01234 012345 0123456

1

2 3 4

7

6

2 2 2 2

1

l2

2 2 1 1

2 2

I

l2

12

l2

48 240 120 8 336 12 42 6 3 3

8 0 10

6 14 14 1 13 7 8

15 1 3 0 1 0 0 4 0 0 1 6 1 0 0 9 0 3 0 6 0

~

~~~

6

2 2 2 2

No

1

72 120 720

5040

3. 12 15 15

15

6 6 6 6 3 3

3 3 3 3 3 3

1. 1, 1 191, 1 2,292 2,2,2 1, 1, 1 2,2,2

36 63 36 63 37 3.63

24 120

4 5

r r

6 7

T

O

5040

r

0

m

0 O

p r p

r p r

0

0 0 0 I 0

On linked arrays of pairs

Table 6 The known LAPSof order u, 8 S u S 9, and depth f = 4 (mod T, or 71,7 2 ) u = 8,

T=

08 01

12 08

(0,1,2,3,4,5,6,7)(8) _

~~

25

02 03

16

24 36 08

06

u = 9,

TI

04

15 26 37

26 37

04

15

15 26 37

8 9

1 2 1 2

08 01 02 03

15 35

23

08

= (0,4,8)(1,7,6)(2,3,

04

1344 28 576 1728

S),

72

04

37 04 15 26

15 26 37

28

28 36 36

24 4 8 24

24 04 08 56

14 08 56 13

_

_

_

~

~

~

23

02 36 08

= (0,7,2X1,5,8)(3,4,6)

15 04 37 26

26 37

04 15

4 4 4 4

37 26 2 15 04

P P

P P

1 O

1 1

Department of Computer Science University of Toronto Toronto. Canada

References [l] R.C. Bose and T. Shimamoto, Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Amer. Statist. Assoc. 47 (1952) 151-184.

[2] R.K. Brayton, D. Coppersmith and A.J. Hoffman, Self-orthogonal latin squares of all orders n f 2,3,6, Bull. Amer. Math. Soc. 80 (1974) 116-118. [3] P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Repts. Suppl. 10 (1973). [4] J. Denes and A.D. Keedwell, Latin Squares and their Applications (Academic Press, New York, 1974). [ 5 ] K. Heinrich, Self-orthogonal latin squares with self-orthogonal subsquares, Ars Combin. 3 (1977) 251-266. [6] A.J. Hoffman and R.R. Singleton, On Moore graphs of diameters 2 and 3, IBM .I.Res. Develop. 4 (1960) 497-504. [7] C.C. Lindner, The generalized singular direct product for quasigroups, Canad. Math. Bull. 14 (1971) 61-64. 181 R. Mathon, 3class association schemes, Proc. Conf. on Algebraic Aspects of Combinatorics, Congressus Numerantium XI11 (Utilitas Math., Winnipeg, 1975) 12S15S.

R. Mathon

292

191 R. Mathon, The systems o f linked 2-(16,6,2)designs, An Combin. 11 (1981) 131-148. [lo] N.S. Mendelsohn, Latin squares orthogonal to their transposes, J. Comb. Theory Ser. A I 1 (1Y71) 187-189. [11] J . Schonheim, On maxima! systems of k-tuples, Studia Sci. Math. Hungar. 1 (1966) 363-368. [I21 R.M. Wilson, An existence theory for painvise balanced designs 111: Proof of the existence conjectures. J. Combin. Theory Ser. A 18 (1975) 71-79. [I31 R.M. Wilson, Concerning the number of mutually orthogonal latin squares, Discrete Math. 9 (1974) 181-198.

Annals of Discrete Mathematics 15 (1982) 29S304 @ North-Holland Publishing Company

SIMPLE STEINER QUADRUPLE SYSTEMS

E. MENDELSOHN* and K.T. PHELPS** Dedicated to N.S. Mendelsohn on the occasion of his 65rh birthday A block design is sometimes said to be simple if it contains no nontrivial subsystems. Doyen [2] showed that there exists a simple triple system for all possible orders u = 1 or 3 (mod 6). In this paper we consider the analogous problems for Steiner quadruple systems. In particular, several recursive constructions for simple quadruple systems are given thereby establishing that the spectrum for these systems is infinite.

1. Introduction

A Steiner quadruple system of order u (briefly SQS(u)) is a pair (P, p) where P is a v-set and p is a collection of 4-subsets of P (usually called blocks) such that every 3 element subset of P is contained in exactly one block of p. Quadruple systems exist for all u = 2 or 4 mod 6 [4]. If (Q, q ) is an SQS(u) and (P,p ) is an SQS(u), then ( Q 9 ) is a nontrivial subsystem of (P, q ) if and only if Q C P, q C 9 and u 3 8. It is well known that for an S Q S ( u ) to have a subsystem of order u, then 2u s u. Finally we say a quadruple system is simple if it contains no nontrivial subsystems. It seems reasonable that if one would choose a ‘random’ quadruple system of order v it wouId be simple and rigid (that is, n o nontrivial automorphisms). However, until now only a finite number of simple quadruple systems were known to exist. The main purpose of this paper then is to give several recursive constructions for simple quadruple systems thereby establishing that the spectrum for these systems is infinite. Analogous results have been proved for Steiner triple systems; Doyen (21 showed that there exists a simple triple system for all possible orders u = 1 or 3 m o d 6 . Lindner and Rosa [9] have shown that there exists a rigid (or automorphism-free) triple system for all u 2 15, u = 1 or 3 mod 6. Babai [ 11, using a major result on the Van der Waerden permanent conjecture has recently shown that almost all triple systems are rigid. Quackenbush [12] has *Supported by NSERC Grant No. A7681.

* * O n leave from Georgia Institute of Technology, Atlanta, G A 30332. 293

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conjectured that almost all triple systems are simple. With respect to quadruple systems these questions remain open. In fact it even remains t o prove that there exists more than one nonisomorphic quadruple system for each n 3 14, n = 2 or 4 m o d 6 . We feel that the constructions presented here could help resolve these other questions as well.

2. Doubling construction

There is a very well-known doubling construction (see [S]) for quadruple systems which we will modify to produce simple quadruple systems. Construction A. Let ( P I p , l ) and (PI,p 2 )be any two SQS(u) where PIr l Pz = 8. Let F and G where F = {Fl,.. . , Fu-l} and G = { G I , .. . , GU-]} be any two 1-factorizations o f K, on PI and Pz respectively and let a be any permutation on the set { I , 2, . . . , u - I}. Define a collection of quadruples q on Q = PI U P2 as follows: ( 1 ) P I U P 2 C 4. (2) {x, y , a, 6) E q if and only if [x, y] E F, and [a,b] E Gj and ia = j . Then (0,q ) is an SQS(2u).

Following Lindner and Rosa [8] we will denote t h e quadruple system (Q, q ) by [Pi U P2]@,, ~ 2 F,, G, a). Now let us define a fragment as the following 8 block configuration:

and a ‘twisted’ fragment as

Clearly if a quadruple system contains a fragment one can delete these blocks and replace them by those of t h e ‘twisted’ fragment. The resulting collection of blocks is still a quadruple system. This method of modifying quadruple systems

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was introduced by Phelps [ll]in relation to another problem. Also Gibbons [3] used a similar idea in his work on nonisomorphic SQS(16). In this paper we will utilize this procedure to destroy subsystems. First we need the following result.

Lemma 2.1. For all n >2, there exist 1-factorizations of Kh with no sub-lfactorizations of order 4. Proof. For 2 n f 1 mod 3 such a 1-factorization can be constructed from the cyclic group (i.e., GK2,). Moreover for small orders one can easily construct 1-factorizations with no sub-1-factorizations of order 4. We present two well-known recursive constructions of such 1-factorizations in the language of latin squares and quasigroups. A 1-factorization of K,,, is equivalent to a Commutative unipotent quasigroup; naturally sub-1-factorizations correspond t o sub-quasigroups. Let P1,P2be latin squares of order n corresponding t o commutative unipotent quasigroups with no sub-quasigroups of order 4. Let N be any latin square of order n containing no subsquare of order 2 (cf. [6,8, 101, etc.) and let fl be its transpose. Assuming that the entries for N and those for PI (and P2) come from disjoint n-sets then it is easy t o see that the latin square (Fig. 1) containing N, NT, Pl,P2 as subsquares is the multiplication table for a commutative unipotent quasigroup of order 2n which contains no sub-l-factorizations of order 4.

Fig. 1.

Next we give a second (similar) construction of commutative unipotent quasigroup of order 2 n - 2 for one of order n. Note that a commutative unipotent quasigroup of order n is equivalent to a commutative idempotent quasigroup of order n - 1. Let P, Q be latin squares of order n - 1 which correspond t o commutative idempotent quasigroups (which in turn are equivalent to the commutative unipotent quasigroups of the previous construction). Let N be a latin square of order n - 1 which contains no subsquares of order 2 and let I P be its transpose. Assuming again that N and P,Q are defined on disjoint sets then we construct a commutative latin square of order 2n - 2 as before. Now choose

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an arbitrary element. say 0, that occurs in N and let us assume that this element lies on the main diagonal of N. Now we replace elements on the main diagonal of Pi and P i with 0 and replace the main diagonal of N and NTwith the main diagonal of Pi(which we can assume is the same as that of P2). The result is a commutative unipotent latin square of order 2n - 2 (see Fig. 2) as desired. It is straightforward to see that it contains no sub-quasigroup of order 4.

Fig. 2.

We are now ready to state and prove the main result of this section.

Theorem 2.2. If there exists a simple SQS(u), then there exists a simple SQS(2u).

Proof. Given a simple SQS(u) (Pl,pl)and another (possibly isomorphic) one ( P 2 , p 2 )o n disjoint sets PI.Pz.one applies Construction A choosing the 1factorization G, so that it has n o sub-1-factorization of order 4. The result is a SQS(2u) whose o n l y subsystems are (PI,pl) and (P2, p2). Note that to have a subsystem of order 8 which intersects each of (PI,p I ) and (P2,p2) in a block, both I-factorizations. F and G, must have sub-1-factorizations of order 4which they do not have. Any larger subsystem must intersect either (Pl,pl)or (Pz, p2) in more than 4 points, but since this intersection is itself a subsystem this contradicts the fact the (Pi, pi)are simple. We can assume without loss of generality that { x , y , z , w }E pl and {a, b, c, d } E p2 and that moreover one can find a permutation a,and 1-factorization F so that t h e remaining six blocks of the fragment will also occur in (a,9). This fragment can now be deleted and replaced with a ‘twisted’ fragment. Needless to say this destroys the subsystems (PI,pl) and (Pz, pz). We claim that no new subsystems are introduced. Clearly any new subsystem, (S, s), must intersect this 8-subset in at least 3 and at most X points; but this means it must contain either one block of the fragment or the whole twisted fragment. The latter case is obviously impossible. Since every block of the ‘twisted’ fragment contains 3 points of P, and 1 point of Pi+l we can assume that {x. y, z , a}E s; but then every triple of S fl P2is contained in a

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block of p2. This means that ( S n P , , s np2) is a subsystem of ( P z , p 2 k a contradiction unless (Sr l P2(d 4. However if S f l Pz= { a , ,a2,a3,a}, then each triple {x, y, ai}, i = 1,2,3, must be contained in a block of the form {x, y, c, d } where c, d E P2 (and of course x, y E PI) but this is also impossible. Similar arguments show that IS rl Pl(< 4 also leads to a contradiction. Hence we conclude that the modified SQS(2u), (0,9*), is simple.

3. Tripling construction Hanani [4] used six constructions in proving that SQS(u) exist for all u = 2 or 4 mod 6. Recently Hartman [5] simplified and generalized three of Hanani’s original constructions. It is Hartman’s presentation of the Hanani constructions which we will use. Although the constructions can be presented as one general construction we will present each variation separately as this will help clarify later arguments. Let L ( N )= {[x, y ] ( ( x - y l = 2j (mod m ) , j E N } be a set of edges of K,,, and let E = K,,,\L(N) be the complement of L ( N ) in K,,,. Then clearly E = {[x, y ] I ( x - yI =j mod m, j # 2k, k EN}. Assuming that m is even, Hartman [ 5 ] showed that E can always be 1-factored.

Construction B.1. Let Xi = {0,1,2, . . . ,6n + 1) x {i} and Pi = {A, B}U Xi. Let (fi,pi), i = 0, 1,2, be any three SQS(6n + 4), where 6n + 4 = 12k + 4 . Let No = G ( O < j d k or 3 k < j d 4 k } , N 1 = C j ( k < j d 3 k } , and N 2 = G 1 4 k < j d 6 k } . Let (Xi, Ei), i = 0, 1,2, be graphs where Ei = { [ x i , yi] 1 [x, y ] L(N,), and x f y } (assuming m = 6n 2). Let Gi, G: be any l-factorizations of (Xi,Ei). Then we have the following construction of a quadruple system (Q, q ) on Q = Po U Pi U P2 [4, 51: (1) pi C q, i = 0, 1,2; (2) (a) {A, xo, y l , zz} E q where x + y + z = 0 (mod 6n + 2); (b) {B,xo, y l , z2}E q where x + y + z 3n + 1 (mod 6n + 2); (3) (a) {xo, To, y l , z 3 E q where x + y + z = - j and T + y + z = j (mod6n + 2) for each j E No; (b) {xo, y l , jjl, z d E q where x + y + z = - j and x + j j + z = j (mod6n +2) for each j E Nl; (c) {xo, yt, 22, Z2)E q where x + y + z = - j and x + y + Z =j (mod6n + 2) for each j E N2; (4) {ai, bi, c ~ +di+l} ~ , E 9 for i = 0, 1,2 (mod 3) where [a, b ] E FkE Gi and [c, d ] E 6 E G:+l and kcvi = j where ai is any permutation of the subscripts of the l-factors. (Note G: can be identical to Gi.)

+

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Construction B.2. When 6n + 2 = 12k + 8 we need a slight variation of the previous construction (also due t o Hartman [5]). W e list only the changes. Let N o = G l l < j s k + I or 3 k + 2 < < < 4 k + 3 } , N I = G I k + l < j ~ 3 k } , N2= (j14k + 3 < j c 6k + 3). Let Ei= {[x, y]II y - XI 12j1, j E N, and I y - X I f 1) for i = 1, 2 and let Eo be defined as before. (3)’(b)’ {xo, y,, y,, z 3 E q where x + y + z - j and x + 1 + z =j (mod 6n + 2) f o r j E N,or where x + y + z =3k + 1 and x + 1 + z =3k + 2 (mod 6n + 2). (c)’ {xo, y,, z 2 ,Z2) E q where x + y + z = - j and x + y + Z = j (mod 6n + 2) for j € Nzor x + y + z = 9 k + 7 and x + y + 2 = 9 k + 6 (mod6n +2). Otherwise the construction is the same.

*

Ei)defined in the above conHartman [ 5 ] showed that the graphs (Xi, structions possess 1-factorizations. In the following lemma we give an explicit 1-factorization in order t o facilitate later arguments. Lemma 3.1. The graphs (Xi,Ei), i = 0, 1 , 2 (as defined in C o n s ~ t i o n sB.l and B.2) can be l-factored.

Proof. Let us ignore the subscript i and assume X = ( 0 , l . . . . ,2m - 1). Consider the subgraph (X, So’)) where So’)= {[x, y] / Iy - XI - j mod 2m). This subgraph is the union of disjoint cycles whose length is 2m/gcd(j,2m). If this number is even, then the subgraph can be l-factored; obviously the cycle lengths can be odd only if the difference is even. In this case we consider the subgraph S(2j) U (2j + 1) which can be l-factored: take cycles of the form C, = (0, Zj, 4j, . . . ,-2j, 1, 1 - 2j, 1 - 4j, . . . , 1+ 2j, 0) and the translates Co+ 2k for k = 0,1,2, . . , ,j - 1, (these cycles always have even length and hence give rise to a pair of l-factors); as the third l-factor take Fo=([2j,4j+ 11, [4j, 6j + 11, . . . , [-4j, 1- 2j], [-2j, 01, [ 1 , 2 j + 11) and its translates Fo + 2k for k = 0, 1,2, . . . ,j - 1. What remains is the fourth l-factor. For the specific case 2) 3 m - 1 (mod 2 m ) we can l-factor the edges of S ( l ) U S(m - 1) instead: take the cycle C = S ( m - 1) U {[O, 11, [ m - 1, m]}\{[O. m - 11, [l, m ] ) of length 2m and also the 2m-cycle S ( l ) U S(m - l)\C. Now with the above 1-factorizations we can show that the following holds.

Corollary 3.2. In the above 1-factorization if the union of any 3 1-factors contains K4, then one of those 1-factors must be S(m), the l-factor with all edges of length m.

Suppose throughout the argument Fl,F2,. . . , Fk are the l-factors constructed in Lemma 3.1. An edge [x,y] is said to have length j if ly-XI - j (mod 2m). There are basically three cases to consider: prod.

Simple Steiner quadruple systems

Case 1. Suppose [x, x + j ] , [ y , y of order 4. This implies that

299

+ j ] E F, belong to some sub-1-factorization

However if 2 edges belong t o the same 1-factor then their respective lengths can differ by at most one. Moreover the smaller length must be even. Using these facts and some simple algebraic manipulations we conclude that j = m (mod 2m). Case 2. Suppose [x, x + 2j] and [ y , y + 2j + 11E F , then this implies: [ x , y ] , [ x + 2j, y + 2j + 11E F,, [ x , y + 2j + 11, [ y , x + 2j] E F,. Again similar arguments show that either 2j = m or (m- 1) (mod 2m); however our 1-factorization (Lemma 3.1) has been constructed so that this is impossible. Case 3. Finally there is one ‘exceptional’ case which can occur only when rn is odd: either edges [0, 11, [ x , x + m - 11 are contained in some sub-l-factorization or [0, m - 11, [ x , x + 13 are. (There are 2 other possibilities actually but the arguments are identical.) Using the specific 1-factorization from Lemma 3.1 plus the fact that this case can only occur when m is odd, one can show by simple but tedious arguments that neither of these cases are possible. In the previous section we constructed 1-factorizations of Kz, with no sub-1-factorization of order 4 to use in Construction A. Our intention is to use the 1-factorizations from Lemma 3.1 in Constructions B (and C). These 1-factorizations can contain sub-1-factorizations of order 4 but any such triple of 1-factors must include the 1-factor with edges of length m. Now in Constructions B (and C) besides being able to choose arbitrary 1-factorizations of the appropriate graphs, one can also choose arbitrary permutations, ai, to match up the 1-factors. We must insure that if the union of three 1-factors of (Xi,Ei) contain K4 then the 1-factors which ai matches them to do not contain K4 in their union. To this end, we claim that each 1-factorization will contain one 1-factor, 6, such that the union 6 U F, U F, will never contain K4 for any choice of I-factors F, fi. More formally, let FI” be the 1-factor of ( X , Ei ) whose edges have length m (mod 2m). Let FY) be the 1-factor all whose edges have length j so that (i,2 m ) is even and moreover m - j = 2h (mod 2 m ) for some h E N,. (Ni is as in Construction B.) If F\‘) U F f ’ U F? contains K4 then FY) must have edges of length m - j but by our definition of ( X i Ei ) these edges are missing. Now choose ai so that ai : FY)+ F$+’)and ai : F$)+ FY+’).It is easy to see that ai will have the desired property. For example in Construction B.l when 2m = 12k + 2 choose j = 1 for graphs (Xi,Ei), i = O , 1, and j = 4 k + 1 for i = 2 . When 2m = 12k+8 as in Con-

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300

struction B.2 choose j = m - 1 = 6 k + 3 for (Xi,E i ) i = 1.2 and j = 2 for i = 0. Finally if we assume that the 1-factorizations Gi are as above and permutations aiare as determined above then we claim the following.

Lemma 3.3. If (Pi,pi), i = 0, 1, 2, are simple SQS(6n + 4) as defined in Construction B.l (or B.2) and ( Q , q ) is the SQS(18n+8) that results from this consmtction under the previous mentioned assumptions, then the (P,,pi)are the only non-trivial subsystems of (Q, 9). Proof. Since the subsystems ( P i , p i ) are assumed to be simple and any 2 subsystems must intersect in a subsystem we conclude that if there exists a subsystem on the set S C Q then IS n Pi/= 0, 1,2 or 4 for each i, i = 0,1,2. Case 1 . S = {A.B}U {x, y } U {x', y ' } U{x", y"} and S n Po = {A,B,x, y } , S n PI = {A,B, x', y'}, S r l Pz= {A,B,x", y"}. We can assume without loss of generality that the following blocks belong t o this subsystem: {A.x, x', x"} , {A.x. y', y"}

.

{ B x y' x"} , { B x x ' y"} ,

{A. y. x'. y"},

{ B y . x', x"} ,

{ A ,y, y ' , x"} ,

{ B y y' y"} .

In general this gives us 8 equations in six variable ( e g , x + x ' S x" = i,

x + y'+ x " = j (mod 2m)). For these equations to be consistent we need i = j + rn (mod 2m). In this case solving these equations gives us that x - y = X'-Y"X''y" = m (mod 2m). In our construction m = 3n + 1. However this implies that [x, y, x', y'] must be a block of 4 but by our choice of a. it cannot

be. Hence no such subsystem can exist; t h e I-factors containing edges of length m are not matched up. Case 2. S f Po l= {A,x, y, z}, S f l PI= {A.x', y', z'} and S n P2 = {A,n } and thus IS1 = 8. This is impossible because in a Steiner triple system of order 7 any 2 blocks must intersect. Case 3. S n Po = {A,x, y, z } , S n PI = {A,x'. y', z'} and S n P2 = { A ,x", y", z"}. We can assume that the other blocks of our subsystem are { A ,x. x'. x " } ,

{A.x, z'. y " } ,

{ A ,x. y'. z " } .

{ A ,y, y ' , z"}

{ A ,y , x', z") ,

{ A ,y, z ' , x"} ,

{ A ,z , v ' , x"} ,

{ A ,z. x', y"} .

.

{ A ,z, z ' , 2").

This gives us 9 equations (e.g. x + x' + x" = i (mod 6n + 2)). Solving these we conclude that x ' = y' = 2' which is impossible. Note that if we were working (mod 6 n ) . then we would conclude that {x, y. z } = {0,2n,4n).

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Case 4. S f Po l= {x, y , z, w}, S n Pl = {x’, y ‘ , z’, w‘}, S f l P2= {r, s}. By a simple counting argument, we can assume that {x, y, w’, r} and {x, y , u’,z’} are blocks of this subsystem. But by our constructions the pair [ x , y ] will not occur in both blocks of type 3(a) and 4. Case 5 . S fl PO= {x, y , z, w} and S fl P1= {R, 8, 2, a}.As before, this will give us a subsystem if and only if both 1-factorizations Go and GI contain sub-lfactorizations and moreover the 1-factors containing these sub-1-factorizations are matched by the permutation a. which is impossible by choice of no. Finally all other cases not explicitly mentioned can be handled in a similar fashion. We are now ready for the main theorem of this section.

Theorem 3.4. If there exists a simple SQS(6n SQS(18n + 8).

+ 4),

then there exists u simple

Proof. In Lemma 3.3 we proved that, given simple SQS(6n+4), one could construct a simple SQS(18n + 8) containing only 3 simple subsystems. One can destroy subsystems (Po,po) and (PI,p l ) in a manner similar t o that of Theorem 2.2 so that no new subsystems are created. The argument is almost t h e same. Again applying this technique to the partial subsystem that remains of (Pl,p l ) and the subsystem (P2, p2) and an argument similar to that of Theorem 2.2 one can conclude that the resulting system contains no subsystems.

4. Second tripling construction

There is a second tripling construction also due to Hanani and generalized by Hartman, which can also be modified to produce simple quadruple systems. In fact the arguments for this result are almost identical to those of Theorem 3.4. Again let Gi be 1-factorizations of graphs (Xi,E i ) where Ei= { [ x i , y i ] 1 y - xf 2j (mod 12k + 4) and j E N.}for i = 0, 1,2. Choose No = GlO<jSk or 3 k + l < j S 4 k + l ) , N 1 = G I k < j e 3 k } and N2= 1 4k + 1 < j S 6k + 1). Now we have the following two constructions (cf. [4,51). Construction C.1. Let (PI, p i ) be an SQS(12k + 8) and Pi = {A,B,C, D } U {0,1, . . . , 12k + 3) x { i } . Construct a quadruple system on Q = PI U PzU P3 as follows: (1) p 1U p2 Up3 C 4 where {A, B,C, D } = p1fl p2 fl p3.

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(2) (a) {A. x(l, Y I , ZZ},x + y + z 0; (b) {B,XO, yi. z?}, x + y + z 3 3k + 1; (c) {C X(J, y i , t 2 } , x + y + z 6k + 2; (d) {D,XU. y!, t ~ x) + , y + z =9k + 3. (Assume all congruences are mod 12k + 4 unless otherwise stated.) (3) (a) {x,). Xo,y l , z2}. x + y + z - j and X + y + z = - J for j E No; (b) {x,,, y l , y,, z 2 } ,x’ + y + z - j and x + + ; z = - j for j E N , ; (c) {xu, y l , z2,Y2}, x + y + z = j and x + y + 2 = - j for j E N?. (4) {al,b,, c,+I,d,+,} where [a, b ]E Fk’E GI and [c, d ] E F$!(r,,E G,+, where t h e subscripts are reduced mod 3. Construction C.2. Let us assume (P,,pi), i = 0, 1, 2, are SQS(12k + lo), then one can construct a quadruple system on Q = Po U PI U P2 just as before except now all congruences are mod 12k + 10. The only changes are No= (j I O < j S 2 k or 3k

+2<j ~

4 +k3},

N I = G I k < j S 3 k + I},

and

N z = 0’ I 4k

+ 3 < j < 6k + 4)

{A.XO, y l , z?}E 4

(2)’

if x + y

+ z =0 ,

{B,x*, y 1 . z z ) E q i f x + y + z = 3 k + 2 ,

{C.XO, y ! , 2 2 } € 4 {D, XO, y t , 2 2 ) E 4

if x + y + z =6k + 5 , if x + y

(mod 12k + 10).

+ z = 9k + 8 ,

We claim that Gi and ai can be constructed as in the previous section (see Corollary 3.2, etc.) and hence we have a result essentially the same as Lemma 3.3.

Theorem 4.1. I f in the above construction the (Pi,p i ) are simple and the Giand ai are as defined previously, then these will be the only subsystem. Proof. Again as in Lemma 3.3 assuming a subsystem exists o n the set S, then 1s n Pi/ = 0, 1.2. or 4 for i = 0, 1.2. Case 1. IS flP;(= 4. S contains two of {A, B,C, D } . The proof is identical to Case 1 of Lemma 3.3. Case 2. I S n Po(= 3, ISnPII= 3. ISnP,I = 2 and S contains one of (A, B, C. D ) . Again this is the same as in Lemma 3.3. Case 3. IS c7 P,/ = 4 and S contains one of {A,B,C, D}. Identical t o Case 3 of Lemma 3.3.

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If S contains none of {A, B, C, D}, then two cases remain. However in both cases the argument is similar to Cases 4 and 5 of Lemma 3.3.

Theorem 4.2. If there exists a simple SQS(6n SQS(18n + 16).

+ 8),

then there exists a simple

Proof. The proof is the same as in Theorem 3.4. In closing this section we remark that other recursive constructions (cf. [4]) could be modified in a similar fashion to produce simple quadruple systems. However there is at least one of Hanani’s constructions which cannot be modified and thus a complete solution of the problem must wait on new constructions of quadruple systems.

5. Conclusion Although the previous constructions yield an infinite class of simple quadruple systems, it is not a linear class and hence does not have positive density with respect t o the spectrum for Steiner quadruple systems. One would like to show that simple quadruple systems exist for all n = 2 or 4 (mod 6); however failing this the next step would be to show that the spectrum has positive density. We remark that this question can be asked about other Steiner systems as well. We can show, for example, that the S ( 3 , 4 + 1,4 3 + l), 4 an odd prime power, having the projective general linear group PGL(2, q3) as its automorphism group, will always be simple. However it appears that an entirely different approach is needed in such a case. Mathematics Department University of Toronto Toronto. Canada

References [l] L. Babai, Almost all Steiner triple systems are asymmetric, Ann. Discrete Math., to appear. [2] J. Doyen, Sur la structure de certains systtmes triples de Steiner, Math. Z. 111 (1%9) 289-300. 131 P.B. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. Thesis, University of Toronto, 1976. [4] H. Hanani, O n quadruple systems, Canad. J. Math. 12 (1960) 145157. [5] A. Hartman, Tripling quadruple systems, Ars Combin. 10 (1980) 255-309. [6] A. Kotzig, C.C. Lindner and A. Rosa, Latin squares with no subsquares of order two and disjoint Steiner triple systems, Utilitas Math. 7 (1975) 287-294.

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[7]A. Kotzig and J. Turgeon. On certain constructions for latin squares with no latin subsquare of order 2, Discrete Math. 16 (1976)263-270. [8] C.C. Lindner and A . Rosa, Steiner quadruple systems, Discrete Math. 22 (1978)147-181. [Y] C.C. Lindner and A. Rosa, On the existence of automorphism free Steiner triple systems, J. Algebra 34 (1975)4 W 3 . [lo]M. MacLeish, On the existence of latin squares with no subsquares of order 2, Utilitas Math. 8 (1975)41-53, [ 111 K.T.Phelps. Some derived Steiner triple systems, Discrete Math. 16 (1976)34S352. [ 121 R. Quackenbush. Algebraic speculations about Steiner systems, Ann. Discrete Math., to appear.

Annals of Discrete Mathematics 15 (1982) 305-318

0 North-Holland Publishing Company

RECTAGRAPHS, DIAGRAMS, AND SUZUKI’S SPORADIC SIMPLE GROUP A. NEUMAIER Dedicated to N.S. Mendelsohn on the occasion of his 65th birthday We investigate incidence structures with diagram-

+ 1.

2

2

--

a

o 2

-

2

. 2

-O

k-1

of rank

For n = 2, examples can be constructed from biplanes, semibiplanes. and strongly regular graphs with A =0, j~ = 2. For n > 2 , there are some examples related t o the construction of a strongly regular graph on 1782 points by Suzuki. We also give some general theorems on diagrams which, e.g., imply that an incidence structure with diagram n

O L 1 ?

i . o;an example of this gives rise to another incidence structure with diagram0 2 2 2 situation can be obtained from the rank 4 representation of PSUj(9) on 36 points.

1. Rectagraphs

All our graphs are finite, undirected, without loops or multiple edges. The nezghbourhood of a set C of vertices (of a vertex x ) of a graph r is the set r ( C ) (resp. f ( x ) ) of all vertices adjacent with all vertices of C (resp. with x ) . f is regular of ualency k if every vertex is adjacent with exactly k other vertices. Often a vertex is simply called a point A n s-claw is a pair (x, S ) where x is a point, and S is a set of s points adjacent with x such that n o two points of S are adjacent. An n-clique is a set of n pairwise adjacent points. A rectagraph (cf. [8]) is a connected, triangle-free graph with the property that each 2-claw is in a unique quadrangle. Perkel [8] shows that a rectagraph is always regular. Examples. (1) Strongly regular graphs with parameters A = 0, p = 2 are rectagraphs. By well-known parameter conditions (see, e.g., [7]) the valency must be of the form k = s2+ 1, with an integer s not divisible by 4, and the number of vertices is then u = :(s’+ s + 2)(s2- s + 2). Examples are known for s = 1 (quadrangle, k = 2, u = 4), for s = 2 (Clebsch graph, k = 5, u = 16) and for s = 3 (Gewirtz graph, k = 10, u = 56) (see, e.g., [7]). By considering the set of points at 305

306

A. Neumaier

distance S 2 from a given point it is easy to see that a rectagraph f of valency k contains u 2 1 + k + (:)points. with equality iff f has diameter 2 iff f is a strongly regular graph with parameters A = 0, p = 2. These graphs also arise in connection with algebraic varieties in projective spaces (see [ 11). (3) The k-dimensional cubes (and, for k 2 5 , the half-cubes obtained by identifying antipodal vertices) are rectagraphs of valency k. (3) A biplane (semibiplane) is a design with the property that any two distinct points are in 2 (0 or 2) blocks, and any two distinct blocks have 2 (0 or 2 ) common points. The incidence graph of a biplane or semibiplane 53,i.e.. the graph whose vertices are the points and blocks of 93, adjacent iff they are incident, is a rectagraph whose valency equals the block size. It is easy to see that a rectagraph is bipartite iff it is the incidence graph of a semibiplane. Also, a bipartite rectagraph comes from a biplane iff its diameter is 3. Proposition 1. In a recfagraph, if x, y are points at distance i, then there are at least i poinfs adjacent to y which have distance i - 1 from x.

Prod. This is true for i = 1, so assume that it is true for i = 1,. . . ,j . If x, y are at distance j + 1 then there is a chain x = xo,xI,. . . = y of adjacent points. By induction, there are at least j points sl,. . . , sj adjacent to xi which have distance j - 1 from x. Now define to = xi, and define fr as the fourth point on the quadrangle containing the 2-claw slxixj+l ( 1 = 1, . . . , j ) . Then to, t l , . . . , fj are j + 1 distinct points adjacent to x,,~ and at distance j from x. This completes t h e induction.

.

Proposition 2. A rectagraph r of valency k has diameter S k , and contains v =s2' points. Moreover, f has diameter k i f f v = 2' i f f f is a k-dimensional cube.

Proof. By Proposition 1, the diameter is S k. If u, is the number of points at distance i from a given vertex x then uo = 1. If we count in two ways the number N, of edges yz with y at distance i, z at distance i + 1 from x, we obtain V , + ~ ( Z + I ) s N, s ui(k - i ) . Hence, by induction, vi =G and u =zZ (t) = 2'. Now suppose that the diameter of r is k. Choose two points a, b at distance k. and define fo as the set of points whose distances to a and b sum up to k. If x E fohas distance i from a and k - i from b then by Proposition 1, there are at least i vertices at distance i - 1 from a, and at least k - i vertices at distance k - i - 1 from b. hence at distance i + 1 from a, which are adjacent with x. But the valency is k, whence there are exactly i vertices at distance i - 1, and k - i vertices at distance i + 1 from a in the neighbourhood of n, and so T ( x )is in r,. Since a E foand f is connected we now have To= f,and the above counting argument gives t)i+l(i+ 1) = Ni = vi(k - i ) , or vi = Hence v = 2'.

(t),

(t).

Rectagraphs, diagrams, and Suzuki's sporadic simple group

307

(t)

Finally, if u = zk then ui= for all i, in particular Uk = 1. Hence for every point a there is a point at distance k from a. In particular, by the above arguments, Proposition 1 holds with 'at least' replaced by 'exactly'. Now fix a E f,and identify x E r (at distance i from a ) with the characteristic vector c ( x ) of the set of all y E r ( a ) at distance i - 1 from x. Then c ( x ) has i entries, and it is easy to show that c is an isomorphism from r into the k-dimensional cube. Remark. This result has been obtained for the incidence graph of a semibiplane by Wild [lo] with a similar proof. Wild also shows that a bipartite rectagraph f of valency k contains u 3 k 2 - k + 2 vertices, with equality iff f is the incidence graph of a biplane. Proposition 3. Let f be a rectagraph on X with valency k. Then the graph I', whose vertices are the pairs (x, a ) with x E X , a = 21, such that (x, a ) and (x', a') are adjacent iff either x = x' and aa' = - 1, or xl is adjacent with x and aa'= E, is a rectagraph of valency k + 1, for E = 21.

Proof. Straightforward. Remarks. (1) If f is bipartite, with classes c",a = 5 1 , then f+is bipartite, with classes {(x, a)I x E c", a = +l}, {(x, -a)I x E c", a = 21). Hence this construction generalizes Wild's doubling construction [101 for semibiplanes. If f is a k-dimensional cube then f+ is a (k + 1)-dimensional cube. (2) f- is always bipartite, with classes {(x, a)1 x E X} for a = +l. If f is strongly regular, then r- is the incidence graph of a biplane with a null polarity. This is a special case of a construction of symmetric 2-designs with a null polarity from strongly regular graphs with h = p - 2 (see, e.g., [7]). If f is a k-dimensional cube, then, again, f- is a (k + 1)-dimensional cube. If f is a half-cube, then f- is a cube with additional edges joining antipodal pairs of points. (3) In the graph obtained from f- by deleting the edges (x, l), (x, -l), each 2-claw is in a unique quadrangle. f! is bipartite, and, unless f itself is bipartite, connected, whence a rectagraph. In particular, every rectagraph is contained in some bipartite rectagraph of the same valency, and a rectagraph of valency k which is not bipartite has at most 2kf' vertices.

2. Magrsms

For the remainder, we assume the reader to be familiar with the paper "Diagrams for geometries and groups" by Buekenhout [2]. We take a slightly

A. Neum aier

308

more general point of view, and call an incidence structure, which is connected under the incidence relation and satisfies Buekenhout’s axiom (l), a geometry. We refine the information contained in a diagram as follows. For a geometry (S, I, A, d ) , suppose that for all flags F of S of type A - {i}, the cardinality of R ( F ) is a constant p, depending only on i E A. Then a labelled diagram of S consists of a diagram of S, together with the label p, attached to node i of the diagram, for all i E A. This also makes clear what we understand by a partially labelled diagram. Since n o confusion is possible we call a (partially) labelled diagram simply a diagram. For example, ; is the diagram for a geometry of rank 1 with p points, y y is an ordinary n-gon, A is the diagram for a complete graph with 2 P p + 1 points, is the diagram for a 2-(1+ r(k - l), k, 1)-design, is the diagram for a Idimensional polar space over GF(9) with q t l q+l 3+1 s = 1, qln, 9. 9, 9”’. 9’ for C?;(q), H5(9), S5(9), Q,49), H49), and W 9 ) , respectively (see, e.g., [3]). Thus the labelled diagram determines the geometry often almost upto isomorphy. Labels for the diagrams of sporadic groups show at once which geometries = o5 5 3 involves H5(4), HS are involved: FZ4with diagram -* involves PG(2,4), JI with diagram ;(6) 7 c ; with diagram o ;

- e;

--

_Q=O

5

involves the ordinary hexagon, whereas HJ with diagram C o (6) o in3 volves the generalized hexagon from G’(2). Sometimes, labels give information even for unlabelled diagrams. For example, since A is always *L o,-+ must be a o and so 2 2 the generalized quadrangle involved must be a rectangular grid (every point on two lines only); similarly, A must be a Note also that is the same as 1 . Th?e ordinary quadrangle has diagram yi’ hence rectagraphs of valency k belong to a diagram ‘2 I. So, for example, the Gewirtz graph yields a diagram v o for the simple group PSL(3,4).Semibiplanes with block size

-

mi.

2

2

g c

k have diagrams 7 (varieties are points, pairs of points on a block, and k? I blocks), and the implication given in Example 3 of Section 1, is a special case of the following theorem.

1

Theorem 4. (i) If there is a geometry with diagram

s nodes

1

O

K

Rectagraphs, diagrams, and Suzuki's sporadic simple group

309

then there also is a geometry with diagram

This geometry has an s-partite point graph, and conversely, if a geometry with diagram (2) has an s-partite point graph, then there is a geometry with diagram (1). (ii) If there is a geometry with diagram

then there also is a geometry with diagram

This geometry has a bipartite point graph, and conversely, if a geometry with diagram (4) has a bipartite point graph, then there is a geometry with diagram (3). Proof. (i) Call the varieties at the s distinguished nodes points more specifically 1-points, . . . , s-points. Replace these varieties by new varieties which we call 1-spaces, . . . , s-spaces. A n i-space is simply a flag consisting of i distinct points. Incidence between i-spaces is containment, and incidence between an i-space and an old variety is as before. The residue of an (s - 1)-space has diagram

The residue of a variety belonging to the central node consists of the cliques of a complete s-partite graph, hence has diagram Hence the new geometry has diagram (2). The converse (in the form stated) is straightforward. (ii) is proved in the same way, using the observation that the incidence graph of a generalized n-gon is a generalized 2n-gon with line size 2.

7; - - 7.

Theorem 4 contains various classical examples as special cases:

A. Neumaier

310

It also applies to the (bipartite) tesselation of a plane by regular hexagons:

p2

y - ( L * (bipartite! e 2

1

2

9

and to a diagram for Held’s group (see [ 111):

Here are two more constructions, the first of which generalizes Proposition 3.

Proposition 5. (i) Every geometry with diagram

E2

Q . . . & C -

2

&-I

gives rise to a geometry with the same diagram and bipartite point graph, and hence to a geometry with diagram

(ii) If there is a geometry with diagram (5), then there is a geometry of the same rank with diagram *&. 2

2

7

--

2

k

(iii) If there is a geometry with diagram (6), then there is a geometry of the same rank with diagram

\. . .A . ?/)

prod. (i) Let S be a geometry with diagram (5) whose point graph is not bipartite. Take as new point set a red and a blue copy of the old point set, and for each i-variety (i > 0), which is an i-dimensional cube on the incident points, define two new i-varieties corresponding to a red-blue and a blue-red colouring of the natural bipartition of the cube. This defines naturally a new geometry with diagram (5). If we separate the red points and the blue points, then, by Theorem 4, we get a geometry with diagram (6). (ii) Define new points (xu, a),where xo is an old point, a = 21. For i > O , define new i-varieties to be either (x,, a),where x, is an old i-variety, a = k l , or (x,-~),where x,. I is an old ( i - 1)-variety. The incidence is given by (xl, a) I (x,, a’)iff x, I x,. a = a’;(x,, a) I (1,)iff x, I x,, i s j ; (x,) I ( x , ) iff x, I 1,.The verification of the diagram is straightforward. (iii) follows from (ii) by Remark 2 of Section 1.

Rectagraphs, diagrams, and Suzuki's sporadic simple group

311

Proposition 6. If there is a geometry with diagram

@A...A, *

(7)

then there also are geometries of smaller rank with diagrams

Proof. Delete the varieties belonging to the deleted nodes of diagram (7) and apply [2, Theorem 71.

Example. By Cameron [6] the group 212Ma acts on the 212 cosets of the binary Golay code as the graph of a parallellism of the 4-subsets of a 24-set. This gives rise to a geometry with diagram= ; '~ 3 the varieties are points, edges, quadrangles, 3-cubes, Ccubes and 8-cubes corresponding to certain 0-, 1-, 2-, 3-, 4-, and 8-dimensional subspaces. Moreover, the point graph is bipartite, whence there is a geometry for the diagram

(the group is 211M24, and the diagram was mentioned in [2, Example 12.31). Truncation according to Proposition 6 gives geometries for the diagrams a==i-.--c-o, 2

2

2

2

2

- - - L a ,

1

2

2

2

2

2

v 2

2

2

0

3

2L and 2L, 2 0 /2

2

21

2

/ *

and then Proposition 5 shows the existence of geometries with diagram 2 /2

L2

k4

for all k 2 2 1 .

20

Finally we mention that the incidence graph of a thick partial plane with the property that any two points on a line are on a unique triangle give rise to a geometry with diagram 4 (') =2 -%, hence to a geometry with diagram

;

3 12

A. Neumaier

If every point is in k lines and every line contains k points, then we have the diagrams

--

(6) ---

and

7

'--*. Unfortunately, I don't know any example of this situation. ,

3. Geometries of type S. For the purpose of the next theorem, call a group H acting on a set X fully 2-transitive if it is 2-transitive and the pointwise stabilizer of two points fixes n o other point.

Theorem 7. Let f,,be a graph with the property that, for some n 2 2, (i) there are n-cliques, and (ii) the neighbourhood of a ( n - 2)-clique is a rectagraph of valency k . Suppose that f,,possesses a group G,, of automorphisms such that (iii) G,, is transitive on n-cliques. and the stabilizer of an n-clique induces on it the symmetric group S,,, and (iv) the stabilizer H of a ( n - 1)-clique C is fully 2-transitive on the neighbowhood of C. If is connected. then it is the point graph of a geometry S of rank n + 1 with diagram

r,,

und G,, is transitiue on the tnaximal flags of S. Let us call an n-clique B of f,,a mate of another n-clique C of r,, if B u C is a K n x Za, complete multipartite graph with n classes of size 2. Let K be a group conjugate to a two-point stabilizer of H, and denote by F,(K) the fixpoint set of K in f,,. To facilitate the induction we also introduce the empty graph r,, and the graph rl with k vertices and n o edge. If f,,,G. satisfy the hypothesis of the theorem, then so d o the neighbourhood rn-l = I',,(x) together with the stabilizer Gn-,= (G,,)xof any point x, with the same group H. Hence the neighbourhood in F,,(K) of a point of F,(K) is a Fn-l(K).T h e essential proo[.

argument is the following lemma.

Lemma 8. For all integers ti one mate in F,,(K).

5 0,

every n-clique conruined in F , ( K ) has exactly

Recragraphs, diagrams, and Suzuki's sporadic simple group

313

Proof. The lemma is trivial for n = 0 and true for n = 1 since, by (iv), F l ( K ) contains exactly two (nonadjacent) points. For n = 2, if Fz(K) contains an edge xy, then it contains, by definition of K, a 2-claw (x, {yy'}) which is in a quadrangle x y x ' y ' , and x'y' is a mate of xy (x' is fixed by K since it is uniquely determined by x, y, y'). If x"y" is another mate of xy then x' and X" are mates of x in the neighbourhood of y, hence x' = x", and similarly y' = y". Hence the mate is unique. Now we proceed by induction, and assume Lemma 8 for 0, 1 , . . . , n - 1 in place of n. For n 3 3 , if F , ( K ) contains an n-clique C, choose three distinct points xi E C,i = 1, 2, 3. Let B be the mate of C' = C- {xI,x2, x3} in the neighbourhood F,-3(K) of x l , x2, x3. Let B,be the mate of C- {xi}in the neighbourhood of xi (i = 1, 2, 3), and let xji be the opposite of xj in €Ii (j# i). Then BI- {xzl, x3I) is a mate of C' in &(K) whence BI = B U {xZlrx31}, and similarly BZ= B U {x32, XIZ}, B3 = B U {xI3,xB}. Therefore, B U {xI2}and B U ( ~ 1 3 ) are mates of C - {xz, x3} in the neighbourhood of x2 and x3, whence xl2= XI3 = yl, say, and similarly x B = = y2, x3I = ~ 3 =2 y 3 . Now it is easy to see that B U{yl, yz, y3} is an n-clique, hence a mate of C,and that this mate is unique. Proof of Theorem 7 (continued). Let C and C' be n-cliques intersecting in a (n - 1)-clique CO,so that C = COU{a}, C' = COU { b } , with two nonadjacent points a, b in the neighbourhood of Co. By (iv) the pointwise stabilizer K of C U C' is conjugate to a two-point stabilizer of H. By Lemma 8, C has a unique mate B in F,(K). For x E C, denote by x' the opposite of x in B U C. An application of Lemma 8 t o rl= Tn(Co), with n = 1, shows that b = a' whence B contains b. Now let B be any mate of C in containing b. For x E C, denote by X the opposite of x in I? U C. Then ii = 6 = a'. If x E C- { a } ,then in the neighbourhood of the ( n - 2 ) d i q u e C - { U , X ) xaxa' , and xax'a' are quadrangles containing the 2-claw (x, {aa'}), whence by (ii), X = x'. Hence I? = B, and C U C' is in the unique Knx2B U C. Hence we have the following:

r,

(P,)

Two n-cliques which intersect in a (n - 1)-clique are in a unique Knxz.

Now i t is easy t o check that we obtain a geometry S of rank n + 1 with the required properties, if we call the ( i + 1)-cliques i-varieties (for i = 0 , . . . , n - l), and the Knx2 n-varieties. Theorem 9. There are geometries with the following groups and diagrams:

A. Newnaicr

314

r,,

r6

Proof. Suzuki [9] constructs strongly regular graphs r,, rs, with 36, 100, 416, and 1782 points and transitive automorphism groups G3 = Aut G2(2), G4= Aut HJ. Gs = Aut G2(4), and G6= Aut Sz, such that the neighbourhood (Gi-,), for i = 6, 5, 4, 3; here r2 (stabilizer) of a point of riis isomorphic to ri-, is a rectagraph with 14 points, namely the incidence graph of the unique biplane with k = 4, and G2= Aut PSL3(2). Hence these graphs satisfy the conditions of Theorem 7 with k = 4, H = S4. Now we consider, in a slightly more general setting, the dual geometries. Let rnbe a graph satisfying (P,,) and let the following hold: Every maximal clique has n points, and every nonmaximal clique is in at least 3 maximal cliques.

(Q,,)

points (0-varieties) and the i-cliques Define a geometry S by calling the KnX2 (n + 1 - 1)-varieties (i = 1 , . . . , n), with natural incidence. This gives a geometry of rank n + 1 with diagram -a----

r,,

2

2

2

.

.o------o 2

2

-

(9)

and can be recovered as the graph on the blocks (n-varieties) defined by calling two blocks adjacent if they are distinct and incident with a common line (1-variety). A geometry arising in this way from a graph rnwith (P,) and (Q,,) is said to be of type S,,. It can be shown that a geometry with diagram (9) is of type S,, iff the block graph satisfies (P,,) and (Q,,). O n the other hand, the dual of the half-cube [2, p. 1281 has rank 3 and diagram (9) with 3 nodes but a complete block graph, hence is not of type S,,. Note that the residue of a block in a geometry of type S,, is a geometry of type s n - , . Let S be a geometry of type S,. For a variety u E S, denote by P(u), B(u) the set of points resp. blocks of S incident with u. Then B(u) is a KnX2 if u is a point. an (n + 1 - +clique if u is an i-variety, i >0, and for every i-clique B of blocks there is a unique (n + 1 - +variety u with Z3( V) = B, and every Knx2 is of the form B ( x ) with a point x.

Reciagraphs, diagrams, and Suzuki's spradic simple group

315

Proposition 10. Let S be a geometry of type S,,, and let u, w be varieties of S. Then (i) P ( u ) c P ( w ) i f f u s w i f f B ( u ) > B ( w ) , (ii) P ( u ) = P(w) iff u = w iffB(u) = B(u).

Proof. Obviously, u s w iff B ( u ) > B ( w ) , and u s w implies P ( u ) c P ( w ) . Assume that U Sw and u is not a point. Then B (u) is a clique not containing B ( w ) . Hence there is a block b € B ( w ) with b e B(u). By (Q,,) there is a n-clique B > B ( u ) with b @ B . By (P,,) and (Q,) there are at least 3 Knx2 containing B, but at most one containing B and 6. Hence there is a KnX2 containing B but not b, and a corresponding point x E S such that b !Z B(x) 2 B. Then B ( u ) G B ( x ) , B(w)EB(x), whence x s u but x j 4 w. Hence P ( v ) P ( w ) , which proves (i). (ii) is a consequence of (i). By Proposition 10 we may identify a variety u with the point set P ( u ) , and we may talk about the intersection of varieties. For the next proposition. call two points adjacent if they are o n a line, and call a triangle a set of 3 mutually adjacent points not contained in a line.

Proposition 11. In a geometry of type S,,, any two adjacent points are on a unique line, and every triangle is in a unique plane. Proof. Let 1, I' be two distinct intersecting lines. Then B(l), B(1') are distinct and there are nonadjacent blocks b E B(l), b' E B(1'). n-subcliques of a KnXz, Let m be the plane with B ( m ) = B(1)- {b}, and let 1" be the line with B(I")= B ( m ) U{b'}. Then B ( m ) B(1) f l B(1") whence 1 and I" are in m. Since m is a* ? - 41,and I" intersect in a unique point. Now if x is a point incident with 1 and 1' then B(x) 2 B(1) UB(1') 2 B(1") whence x is also incident with I". Hence 1 and 1' have a unique point in common, and so two adjacent points xy are on a unique line Xy. Now let xyr be a triangle not in a block. Define B1= Bz = B(;;f), B3 = B(F).Then B1,Bz,B3 are disjoint n-cliques. Since B ( x ) 2 B 1U B2, and contains 2n blocks, B ( x ) = BI U BZ, and similarly B(y) = B1 U B3,B(z) = BzU B3. Now B ( x ) ,B(y), B ( z )are KnxZ whence B = B,U B 2 U B 3 contains 3n blocks, and every block is nonadjacent to exactly two other blocks. If B contains a triple (bo, bl, b2) such that bo is not adjacent with bl and b2, but bl and bz are adjacent, then Cz= {bl,b2}may be extended to a 3-clique C, in at least 3n - 5 ways, C3to a 4-clique C, in at least 3n - 8 ways, etc., and we see that there is a (n + Itclique, violating (Q,,). Hence, if bo is nonadjacent with b, and bZ, then bl and b2 are nonadjacent. Therefore, B is a KnX3. But this contradicts (P,,). Hence every triangle is in a block. Repetition of the argument in the residue of this block, etc.,

c

B(G),

316

A. Neuniaier

shows now that every triangle is in a plane. By part (i) of the next proposition, this plane is unique.

Proposition 12. Let S be a geometry of type S., and suppose that, for some m n. if two varieties are contained in some m -variety, then their intersection is a uariety or empty. (This holds, e.g., for m = 2.) Then the following holds: (i) Two varieties containing a common line intersect in a uariety. (ii) A uariety intersects an i-uariety with i < m in a subspace or empty. Proof. The hypothesis holds for m = 2 since every plane is a J ,, hence a partial plane. (i) Let u, w be varieties containing t h e line 1. Then B ( u ) U B ( w ) is a subset of B(1). hence a clique. Therefore, there is a variety u such that B ( u ) U B ( w )= B ( u ) . and a point x is in u nw iff B ( x ) _ >B ( u ) U B ( w ) = B ( u ) i f f x is in u. Hence u f l w = u. (ii) Let u, w be varieties such that u n w is neither empty nor a variety. We may choose L’, w such that the rank of w is minimal, say. w is an i-variety. Then neither of t’. w is a point, and by (i), u and w have no common line. Hence B ( u ) . B ( w ) are cliques, and B ( u ) U B(w) is not a clique. So there are nonadjacent blocks b E B ( u ) , c E B ( w ) . Let u be the ( i + 1)-variety with B ( u ) = B ( w ) - {c}, and let u’ be the i-variety with B(u’)= B ( u ) U {b}. I f x E u fl w , then B ( x ) 2 B ( u ) U B ( w )2 B(u’)U B ( w ) , whence x E u’ fl w. Therefore u n w c u’ n w. I f u’ r l w = w‘ is a variety, then u fl w’ C u n w = u fl ( u f l w ) C u f l ( u ‘ f l w ) = u n w’ and so u 17 w‘ = u fl w is not a variety. By the minimality of the rank of w. w’ = w, i.e., w s U‘ contrary to our construction. Hence u ’ f l w is not a variety. But u’ and w are contained in the (i + 1)-variety u, hence, for i < m, in some m-variety, contradiction.

Theorem 13. For n = 3, 4, 5, 6, there are geometries with diagram satisfying axiom (3) of [2]. The lines haue 3 points, and if abc, ade, bef are lines, then cdf is a line. 0

3

u

2

c

-

2

-

+

n-l

Roof. By [2, Example 12.1 and Theorem 61 the duals of the geometries of Theorem 9 satisfy the assumptions of Proposition 12 with m = 4. Now apply Proposition 6 to get t h e diagram, [2, Theorem 61 to get axiom (3). and Proposition 11 to get the closure property for the lines.

Remarks. (1) In case of rank n + 1 2 5 , the geometries of Theorem 9 do not

satisfy axiom (3) (see [ 1I]). (2) The geometry 7’ ; f ;for Aut HJ is not self-dual since the residue of a block has 21 points whereas the residue of a block of the dual has 24 points.

Recragraphs, diagrams, and Suzuki’s sporadic simple group

317

There is another realization of the diagram ; 1 ; y , namely a triple cover of the geometry given in Theorem 3 for Aut Gz(2). This is due to the fact that G2(2) has a simple subgroup PSU3(9) of index 2. PSU3(9) has a rank 4 representation on 36 points over PSL3(2), and an associated directed graph r on 36 vertices with in-valency 7 and out-valency 7 (e.g., take as vertices the points, lines, and flags of PG(2,2), and another vertex 0, and take as edges o+p, p+1 [ p e l ] , p + ( p , 1 ) [ P E I ] , I - 4 I + W ’ ) [p=In1’], (pJ+ [ I = pp’], (p, I ) + 1 [p E I ] , (p, I ) + (p’, 1’) [p E I3 p’E I’, p # p’, I # I’). Now take three copies X I ,X 2 , X3of the vertex set o f f , and define the 3-partite graph f $ on XIU X z U X3 with edges iff x + y, i mod 3. This graph f 2 satisfies the conditions of Theorem 7. Hence it leads to a geometry with diagram + o 0 , and since it has a 3-partite point graph, also to a geometry with 2 2 2 3 diagram

by Theorem 4. Suzuki’s graph for Aut G2(2) is obtained by identifying all xl, x2, x3. Notice the similarity with Proposition 5 ! Problems. (1) Is there a geometry extending the sequence of geometries in Theorem 9 (resp. Theorem 13) ? (2) Classify all partial planes with lines containing 3 points such that if abc, ade, bef are lines, then cdf is a line. (3) Are there geometries with diagram

related to HJ,G2(4) and Sz ? (4) Are there any other applications of Theorem 7, with n 3 3 ? A! Af (5) Are there geometries with diagram 0 n o ? For q = 2, is 2 2 4 9+1 the same as*, and we have the above problem and examples. For q a Al prime power we have at least examples of q y l , whose varieties are points and lines (rank 0), non-flags (rank I), and flags (rank 2) of a projective plane of order q, with a suitable incidence defined on the varieties. The details are left to the reader. 4

We close with some remarks on Problem 4. If r, satisfies the conditions of Theorem 7, then the neighbourhood of a (n - 2)clique is a rectagraph with a

31x

A. Neumaier

point- and edge-transitive group G of automorphisms such that G, is fully 2-transitive on T ( x ) . It would be interesting to characterize such rectagraphs. Beside the n-dimensional cubes and t h e half-cubes (obtained by identifying opposite vertices of an n-dimensional cube with n 5 5). which realize H = A, or S,, I know of a number of sporadic examples: The Clebsch and Gewirtz graph (and their bipartite 2-cover, from Remark 3 of Section 1) realize H = A5 ( k = 5. u = 16 or 32) and H = PSL(9) (k = 10, u = 56 or 112), and the incidence graphs of the biplanes B(4). B(S), and B(6) of [4] realize H = S, ( k = 4, u = 14) H = A5 ( k = 5, u = 22), and H = A6 or S6 (k = 6, u = 32). Except for the case H = So. discussed above, it is not known whether any of these extend tea--..+- _' Q

.

Institut fur Angewandte Mathematik Universitat Freiburg Freiburg, W-Germany

References [ I ] A.A. BNen and J.W.P. Hirschfeld. Applications of line geometry over finite fields 11. Geom. Dedicata 7 (1978) 333-353. (21 F. Buekenhout, Diagrams for geometries and groups, J. Combin. Theory Ser. A 27 (1979) 121-151. [3] F. Buekenhout and E. Shult. O n the foundations of polar geometry, Geom. Dedicata 3 (1974) 15-5-17O.

(41 P.J. Cameron, Biplanes, Math. 2. 131 (1973) 85101. [S] P.J. Cameron. Permutation groups with multiply transitive suborbits, Proc. London Math. SIX. 111 Ser. 25 (1Y72) 427-440. 161 P.J. Cameron, Parallelisms of complete designs, London Math. Soc. Lecture Note S e r . 23 (Cambridge Univ. Press, London. 1976). [7] X.Hubaut. Strongly regular graphs, Discrete Math. 13 (1975) 357-381. [8) M. Perkel. Bounding the valency of polygonal graphs with odd girth, Canad. J. Math. 31 (1'379) 1307-1321. [9] M. Suzuki, A simple group of order 448, 345. 497, hoo, in: Theory of Finite Groups (Benjamin. New York, 1969) pp. llS119. [lo] P. Wild, Semibiplanes. to appear. [ l l ] F. Buenkenhout. personal communication.

Annals of Discrete Mathematics 15 (1982) 319-331

@ North-Holland Publishing Company

LOGIC OF EQUALITY IN GEOMETRY

R. PADMANABHAN* To Professor Nathan S.Mendekohn on his 65th birthday We present here a new set of rules of inference which formalize various geometrically motivated techniques of equational derivations in algebras associated with geometries, notably algebraic geometry. This enables us to prove several deduction theorems, known in geometly, purely within the framework of general algebra. We further demonstrate that the classical algebraic systems occurring in geometry, especially quasigroups in projective curves and complete irreducible algebraic varieties of algebraic geometry, do admit our rules of inference. This brings out, among other things, the pure universal algebraic essence of the classical theorem of Chevalley that every algebraic group whose underlying variety is irreducible and complete is necessarily commutative. Using these techniques we characterize those totally symmetric quasigroups which enjoy these rules of derivation. n e s e turn out t o be precisely the cubic quasigroups. This, in turn, gives us a decision procedure to tell when two expressions f and g define the same function over a plane cubic curve where f and g are built from the familiar non-associative binary law of composition of chord-tangent construction.

1. Introduction

The logic of deriving equational identities in mathematical systems is rich and varied. The rules for derivation that one needs in a given context largely depend upon the class of models which one happens to be dealing with. Thus, if the class of all models is an equationally defined class of algebras of a specific type, then, of course, the rules of derivation are completely described by the familiar equational logic (see [3, p. 3811 for these rules). On the other hand, if we have a class of models having a richer structure like, say quasigroups of algebraic curves then we may be able to add some stronger rules of derivation to our basic equational logic. Thus, for example, while not every group is abelian, every algebraic group whose underlying variety is irreducible and complete is necessarily commutative. In order to facilitate the discussion of this phenomenon from a purely equational theoretic point of view, we formulate a set 93 of rules of inference and introduce a corresponding consequence relation: If 2 is a set of identities or implications of type 7 and u is another identity or This research was supported by a grant from NSERC of Canada.

319

320

R . Padmanabhan

implication of the same type, then

means that for every mathematical system \2I whose first order theory satisfies the rules of derivations in 97, Yi 2 implies 91 u,i.e., whenever '?I satisfies 2 for some interpretation of the operation symbols of type T, &I satisfies u for the same interpretation. For sake of brevity, let us call a binary operation m(x, y ) E R defined in ?l = ( A ;R) a 'group law' if there exists a unary operation E R such that the reduct ( A ;m. -') is always a group, i.e., m and satisfy all the requirements for being group operations. With this terminology, the commutativity of a complete algebraic group becomes the following theorem.

+

-'

Theorem 1.1. {rn is a group l a w }

-'

kg { m ( x , y ) = m ( y , x)}.

In what follows we prove a stronger theorem. Further we show that complete algebraic varieties d o enjoy all the rules of derivations in 3. Thus 3 qualifies t o be what might be called the 'geometric logic'. We also characterize those totally symmetric quasigroups admitting the rules % and, not surprisingly, they all turn out to be the so-called cubic quasigroups. For the definitions and properties of various concepts of universal algebra like types of algebras, n-ary polynomials, algebraic functions, free algebras, etc., we refer t h e reader to the relevant sections of [3]. Similarly, we follow the treatment o f Mumford [ 5 . 6 ] and Shafarevich [lo] for the algebrageometric concepts like complete varieties, product varieties, morphisms, etc. Throughout this paper, we shall assume that all our algebraic varieties (algebraic curves) are complete, irreducible and are defined over an algebraically closed field k. Thus, in a pointed variety ( A ; e), e is a k-point and similarly, all morphisms are defined over k. But for the proof of Theorem 3.2 the whole paper is free from any algebraic geometry. In the course of the proof of Theorem 3.2 we list, of course, all the relevant aigebra-geometric results needed.

2. The logic of geometry All the classes of mathematical objects we come across in this paper like groups. quasigroups, vector spaces and algebraic varieties have a well-defined concept o f a direct product (see [S, p. 64 and 1041) for algebraic (complete) varieties. Also all these objects A = ( A ;R) have a well-defined class of

Logic of equality in geometry

32 1

mappings of the product sets A" into A which are built out of their basic structure R. Thus, for universal algebras like groups, rings and quasigroups we have the algebraic functions (i.e., polynomial functions with some variables, possibly none, being substituted by constants, see [3, p. 451) and algebraic varieties over algebraically closed fields have what are known as morphisms (ratios of polynomial functions, see [lo, p. 251). Again, in all these examples, the n-ary maps include the projections and are closed with respect to substitution and the composition of functions. With these examples and their common features as our backdrop, we denote a mathematical system as a triplet 81 = (A;R, 9)where A is the underlying set of a, R provides a structure for A and 9 is a given class of n-ary maps from the product sets A" into A such that (i) 9 contains all the projections, and (ii) 9 is closed w.r.t. both substitution and composition of functions. Thus, for example, an algebraic 9) where the reduct (A; 0)is an group is a mathematical system ?I = (A; 0, algebraic variety with 9 as its morphisms such that for some m, - I E 9 the reduct ( A ;m, -') is a group, in other words, A has a 'group law m' such that both m and the inverse are morphisms. For lack of a better word, we call the elements of 9 as polymorphisms. As customary in algebra, let us reserve the symbols x, y , z for variables and a, b, c for constants. The symbol e always denotes a nullary operation, i.e., a distinguished element like the identity element of a group. Let f and g denote, respectively, polymorphism symbols of arity 1 + m and n respectively with Lmalandna2.Let

Consider now the following two rules of inference: (1)

from 3 u 3 b V x cf(x, u ) = b) infer

(2)

from 3 a V x (g(al,a2,. . . , a i - l , x, a i t l , .. . , a,,)= g ( a t ,a2,. . . , a,) + x = Uj)

and let

V x V y V z cf(x, y ) = f(z, y ) ) ;

infer V a 3 x (g(al, a 2 . .. . , u i - l ,x, a i t l , .. . , a,,)= a i ) % = 8 u {(I), (2))

where 8 is the five familiar rules of equational logic (see [3, p. 3811) where now, we interpret, of course, all the polynomial symbols as polymorphism symbols.

R. Padmanabhan

322

Remark. If an algebra is ‘abelian’ in the sense that (see, e.g., Freese and

McKenzie, The Commutator, Section 5) all its operations are built out of morphisms of an abelian group, save a constant thrown in, i.e.,

.

!(XI, . . . x.) =

crl(xI)+.. . + a . ( x . ) + f ( k )

wheref(k) is a constant element depending only upon f , then it is easy to see that (1) is valid in every abelian variety. On the other hand, as proved in Theorem 3.2 of this paper, (1) is valid in any abelian variety (in fact, in any complete variety) occurring in algebraic geometry. Thus, the property (1) can be viewed on the universal algebraic essence of ‘abelian varieties’ whether they occur in the context of commutators in universal algebra or as algebraic curves or varieties in algebraic geometry.

Definition 2.1. A mathematical system ? l = ( A ; R , 9) is called a geomehic universal algebra if all the rules of derivation in % are valid in the first order theory of the reduct ( A ;P), i.e., the rules of 3 are satisfied for all choices of polymorphisms in 9. Let z‘ be a set of identities of type tYPe.

7

and u be another identity of the same

Desnition 2.2. We say that u is a geometric consequence of 2, in symbols, C kgu,if for every geometric universal algebra ‘21, we have \zI k 2 implies \)I u in the usual sense of satisfiability. This means, of course, that if satisfies C for some interpretation of the polymorphism symbols, then ?I satisfies (+ for t h e same interpretation.

+

3. Statements of the main results In an elegant appendix to [6] Rarnanujam proved that if a binary morphism m of a complete variety merely possessed a two-sided identity, then rn must be a (commutative) group law ! Theorem 1.1, motivated by and patterned after this result, characterizes the (abelian) group subtraction in complete varieties.

Theorem 3.1. { m(x, x ) = e?m (x, e ) = x} kgu where u is any identity valid in the algebra (a ;x - y, 0). with m (x, y ) being interpreted as x - y and e as 0 . Theorem 3.2. Every complete algebraic variety 3 = ( A ;R, S) with 9 as its morphisrns is a geomehic universal afgebra.

Logic of equality in geometry

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Corollary 3.3. (Chevalley-Weil). If the underlying algebraic variety of an algebraic group is complete then the group is commutative.

-

Let us recall that a groupoid % = (A, ) is called a totally symmetric quasigroup if it is commutative and satisfies the identity x(xy) = y (see, e.g., [l]). It was Etherington [2] who first observed that the set of all points on a non-singular plane cubic curve C can be given the structure of a totally symmetric quasigroup where the binary operation x o y is defined to be the unique third point where the chord joining x and y (the tangent at x if y = x ) meets the curve again. Using the classical Bezout theorem and a counting argument, he proved that (C,.)is entropic in the sense that it satisfies the entropic (or medial) identity ( x o y ) 0 ( z o t ) = ( x o z ) ( y o t ) . It is equally well known that the complete variety C-as a smooth irreducible algebraic curve--can be given the structure of an abelian variety (see [lo, p. 1481) and hence (C, 0,9 ) is a geometric universal algebra. Manin, in a series of papers (see, e.g., [7]), investigated such totally symmetric quasigroups occurring naturally in algebraic surfaces of higher dimensions but none of them, except the above planar case, was entropic. Our heuristic reason for such quasigroups being not entropic is that they do not satisfy our geometric rules 3 of inference. This explains the methodological and historical background for our next result. 0

-

Theorem 3.4. A totally symmetric quasigroup 8 = ( A , ,$3’) with 9 as the set of all algebraic functions of ( A ;* ) is a geometric universal algebra iff the algebra (A, ) satisfies all the identities valid in the planar cubic quasigroup ( C ; ).

-

0

4. Proofs

Proof of Theorem 3.1. Let us write the binary operation m(x, y ) simply as juxtaposition xy. Thus we are given that a geometric universal algebra 3 = (A; ,e, 9)satisfies the two identities (3)

xx = e ,

(4)

xe = x

Form the binary polymorphism f(x, y ) = x(xy). Now

f(x, e ) = x(xe) = xx by (4), =e

by (3)

9

R. Padmanabhan

324

and hence, by rule (1) of 9, we infer the validity of the identity f ( x , y ) = f ( r , y ) . Thus,

Hence, '$1 satisfies the identity

Now, form the ternary polymorphism

and compute f ( e , e, z ) = e ( ( z e )( z e ) )= ee =

e

by ( 3 ) , by (4),

and hence, by the rule of inference (I), we have

and it is well known that this identity x ( ( z y ) (zx)) = y characterizes the binary operation of subtraction. x - y , in abelian groups (due to Higman and Neumann, see, e.g., [8]). The proof of the theorem is complete. prod of

Theorem 3.2. Let Yi = ( A ;0,9)be a complete irreducible algebraic variety over an algebraically closed field k with 9 as its set of morphisms. We want to conclude that ?I is a geometric universal algebra. To facilitate an easy rendering of the proof, let us first collect a few basic facts about such varieties. Since all the varieties in question are irreducible we omit this adjective. Fact 1. Let f : X -+ Y be a morphism, with X complete. Then f ( X ) is a complete subvariety of Y [5, p. 1041. Fact 2. If X and Y are complete, then X X Y is complete (cf. above).

325

Logic of equality in geometry

Fact 3 (dimension theorem (e.g., [lo, p. 601)). Iff : X + Y is a morphism of complete varieties, then, for all points y E Y,dimf-'(y) > dim X - dim(1mage f) 3 0. Fact 4 (rigidity lemma, [lo, p. 1521 or [6, p. 431). Let X be a complete variety, Y and Z any varieties and f : X x Y + 2 a morphism such that for some point y o € Y,f ( X X {yo}) is a single point z O EZ. Then there is a morphism g : Y + 2 such that f = g o p z where pz : X X Y ---* Y is the second projection. Let now, for some (rn + n)-ary morphism f of 8 ,

3 a 3 b V x (f(x,a) = b ) be valid. This simply means that for some k-points and for all (xl, . . . , x m ) € A" we have

(al,

. . . , a n ) €A", b E A

and, of course, f : A" x A" + A is now a morphism of the products of the two complete varieties A" and A". Thus, by the rigidity lemma, we have

and thus, V x V y V z f ( x , y) = f(z,y), and this establishes the validity of rule (1). Let now

be true in the complete variety (21. Let us abbreviate the point ( a l , . . . , ui-l, f ( a l ,. . . , a,,),u ~ +. .~. ,, a,) as b. Consider the morphism (p : A" + A n ,defined by

It is clear that (al,.. . , a , ) € (p-'{b}.Let (xl, . . . ,x , ) E (p-'{b}.Then we have

which, in turn, implies that

R. Padrnanabhan

326

and this, by our assumption, forces that xi = ai. Thus cp-'{b)= {(al,. ..,a,,)),a single point. Hence 0 = dim cp-'{b}2 dim A" - dim rp(A")2 0. So dim A" = dim cp(A")and since cp(A")is complete as well, we must have cp(A")= A", i.e., cp is surjective. Thus given an a E A", 3 x E A" such that

which means that 3 xi E A such that f(&, .. .

.

x,, ai + 1, . . . , an)= Q i .

c2-1,

This is precisely rule (2).

Remark. The above discussion clearly demonstrates that the two rules of our geometric logic are simply universal algebraic formulations of the rigidity lemma and the dimension theorem. The central theme of this paper (as well as that of [9]) is to demonstrate that these algebraic versions of the above two crucial properties of complete varieties form a basis for the process of equational deduction involving morphisms in complete varieties. Proof of Corollary 3.3. Let \u = ( A :0, 9)be a complete algebraic group. This simply means that Vl is a complete variety and moreover there is a binary morphism m E 9 ' which is a group law. Now consider t h e binary morphism * : A x A + A defined by x

* y = m(x, y-')

Since m ( x , x - ' ) = e, this is a 0-ary morphism, i.e., e is a k-point of '21 and we have x*e=x.

x*x=e.

Now, by Theorem 3.2, complete varieties are geometric and hence, by Theorem 3.1, the binary morphism * satisfies all the identities valid in the algebra (H ;X - Y,0) of integers. In particular, x * (x * y ) = y, meaning m(x, m ( x , y - ' ) - ' ) = y, i.e., m ( x , m ( y , x - ' ) ) = y. Writing juxtaposition for m, we have x y x - ' = y, or equivalently, x y = yx.

Proof of Theorem 3.4. Recall that a totally symmetric quasigroup is a groupoid satisfying the identities, (see e.g., [l] or [7]), xy=yx

and

x(xy)= y .

Logic of equality in geometry

327

It is immediate that a non-singular geometrically irreducible plane cubic quasigroup satisfies the above two laws (see Fig. 1). Continuing in the spirit of the Mumford-Ramanujam theorem, we give proof of Theorem 3.4 under a much Weaker hypothesis, viz. assuming the above totally symmetric (TS) property for just one variable and one chosen constant element e, thus, (i) e(ex) = x , (iii) xe = e x .

Fig. 1.

Lemma 4.1. TS kga,where u is any identity valid in an irreducible plane cubic quasigroup (C, * ) over an algebraically closed field k .

Proof. Let 91 = (A; * , e, S) be a geometric universal algebra satisfying the

identities TS with, of course, * and e belonging to 9, i.e., * and e being, respectively, a binary and nullary morphism of (21. In the context of algebraic geometry, this simply means that : A X A 3 A is a morphism and e is a k-point. Consider now the binary polymorphism f ( x , y ) : A X A + A defined by

We have

f ( x , e ) = x(ex) = e by TS(ii) , and hence, by rule (1) of 93, f(x, y ) = f(z, y ) V x , y , z E A. In particular, we have f(x, y ) = f(e, y ) which gives us the identity

Consider now the 4-ary polymorphism f : A4+ A defined by

R. Podmanabhan

328

f ( e , e, z, f)= ((te)e)( z t )= t(zt)= z , and so, by rule (1) of 3, f ( z , y, z, t ) is independent of the variable t. Thus f ( x . y. z, r) = f(x, y , z, u ) . In particular, putting u = y, we get the identity (see Fig. 3)

Fig. 2.

Let x = zt in the above. Using the identity (6) we have

and since the left multiplication is, by (6) again, an onto function we get the commutative law xy

=

yx .

Now. following the classical construction of a group law on the k-points of a plane cubic curve (see, e.g., [7, p. 11821 or [lo, p. la]), let us define the binary polymorphism + : A x A + A by the rule x + y = (xy)e

and a unary polymorphism - : A + A by - x = x(ee). It is clear that x + y y + x and x + e = x . Now x

and

+ (-x)

= (x(x(ee)))e =

(ee)e = e

=

Logic of equality in geometry

329

Fin a11y ,

and hence the reduct (A; +, -, e ) is an abelian group and yz = ee - y - z. This completes the proof that 3 = (A; * , e, 9 ) satisfies all the identities valid in the plane cubic quasigroup (C; ) where C is an irreducible non-singular plane cubic curve over an algebraically closed field k (cf. (2), [7, p. 11711). Theorem 3.4 also follows now. There are many natural examples of totally symmetric quasigroups like Steiner triple systems and even the geometrically motivated quasigroups of cubic hypersurfaces of Manin which do not arise out of abelian groups as prescribed by the recipe x 0 y = kx-’y-’ and hence do not admit the geometric rules of derivation Y4 which are patterned after the deduction techniques of algebriac geometry. Thus Theorem 3.4 can be viewed as a universal algebraic characterization of totally symmetric geometric quasigroups. In particular, the ‘word problem’, i.e., the existence of a decision procedure for telling when two polymorphisms f and g in 9 represent the same function, is solvable for totally symmetric geometric quasigroups. Since f and g are both algebraic functions of (A; ), a simple expansion using the above recipe would reveal whether f = g or not. A few examples of such ‘configuration theorems’ are shown in the diagrams in Figs. 2 and 3.

-

Corollary 4.2. The following two identities (8) and (9)form an equational basis for the set of all identities valid in the cubic quasigroup ( C ; . ) where C is an arbitrary non-singular, irreducible plane cubic curve over an algebraically closed field k, where

330

R.Padmanabhan

Prod. From the proof of Theorem 3.4, it is sufficient t o derive the commutative law. Put z = x in (9) and use (8) to get

Multiplying both sides of the above identity on the right by y and using (8) again we get the desired commutative law tx = xt. The identity (9). drawn as a configuration theorem on the points of the cubic curve, becomes the Pappus-Pascal Theorem (see Fig. 3). Hence, Corollary 4.2 can be reformulated as follows.

Fig. 3

Theorem 4.3. The Pappus-Pascal Theorem for the plane cubic curve implies all the configuration theorems which are valid for the points on an arbitrary cubic c u m . The usual Pappus Theorem for the projective plane is, of course, a speciai case of this if we take the cubic as the product of three linear equations. 5. Conclusion

In all the above computational proofs we have made use only of rule (1) and not of rule (2) of 9.In [9] we have made full use of '3 to prove that a complete variety can admit at most one Malcev polymorphism and, in that case, it already becomes an abelian variety. There are other well-known classes of algebras like f i n e and entropic algebras which are geometric in our sense. These applications and other related investigations will be reported in subsequent publications elsewhere. Acknowledgment I would like to thank the Editorial Committee of Discrete Mathematics for inviting me to contribute to this volume honoring my friend and colleague Dr.

Logic of equality in geometry

331

N.S. Mendelsohn who, during the past twelve years, has influenced my thinking a lot in the interrelations among algebra, geometry and combinatorics. I would also like to thank my teacher Professor M. Venkataraman of the University of Hyderabad, India for stressing, time and again, the organic wholeness of mathematics through his innumerable seminars, classroom lectures and many private decussions. Finally, I wish to express my appreciation to all members of G. Gratzer’s universal algebra seminar here at the University of Manitoba, for all their constructive criticisms and especially to Bob Quackenbush for some corrections and suggestions of clarifying nature. Department of Mathematics University of Manitoba Winnipeg, Canada

References [I] R.H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc. 55 (1944) 19-52. [2] I.M.H. Etherington, Quasigroups and cubic curves, Roc. Edinburgh Math. Soc. 14 (1%5) 273-291. [3] G. Gratzer, Universal Algebra (Springer, New York, 1979, 2nd editon). [4] S. Lang, Introduction to Algebraic Geometry (Interscience, London, 1958). [5] D. Mumford, Introduction to algebraic geometry, Mimeographed Notes, Harvard University. (61 D. Mumford, Abelian varieties, Tata Instit. Lecture Notes (Oxford Univ. Press, London, 1970). [7] Yu.1. Manin, Cubic hypersurfaces I. Quasigroups of classes of points, Math. USSR-Izv. 2 (1%8) 1171-1191. [8] R. Padmanabhan, Single equational axiom systems for abelian groups, J. Austr. Math. S o c . 9 (1969) 143-152. [9] R. Padmanabhan, Uniqueness of Malcev polynomial in complete varieties, Abstract #80TA146, Notices A M S (1980) 475. [lo] I.R. Shafarevich, Basic Algebraic Geometry (Springer, New York, 1977).

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Annals of Discrete Mathematics 15 (1982) 33>340 @ North-Holland Publishing Company

ON AXIAL AUTOMORPHISMS OF SYMMETRIC DESIGNS Fred PIPER Dedicated to N.S. Mendekohn on the occasion of his 65th birthday

This is a short survey article to provide the background material for current research into symmetric designs with axial automorphisms. No proofs are given. The aim is merely to motivate a particular type of problem and state the progress which has been made.

1. Introduction

A collineation a of a finite projective plane P is called axial if it fixes each point on a line 1 and central if it fixes each line through a point P. (The line I is called an axis of a and P is a centre.) Axial collineations have played a central role in the study of projective planes. The following result gives two simple, but fundamentally important, properties of axial collineations. Result 1. Let a be a collineation of a finite projective plane. If a # 1, then (a) a has at most one axis and at most one centre, and (b) a has an axis if and only if a has a centre.

If a has axis 1 and centre P, then a is an elation if P E I and a homology if P 1. It is a consequence of Result 1 that every non-identity axial collineation of a finite projective plane is either an elation or a homology. Another consequence is that the only points fixed by a, or any non-identity power of a, are P itself and those on 1. This observation gives the following result. Result 2. Let a # 1 be an axial collineation of a finite projective plane of order n. Then (a) a is a homology if and only if la1 1 n - 1 and (b) a is an elation if and only if la1 I n. Furthermore the elations with a given axis form a group whose order divides nz. Probably the most important concept associated with axial collineations of projective planes is that of (P, 1)-transitivity. A projective plane P is said to be (P, 1)-transitive if, for any pair of distinct points A, B with P f A, P f B, A @1, 333

334

F. P i p

BfZ 1 and P A = PB there exists a collineation a with centre P and axis 1 such that A" = B. The importance of this concept is illustrated by the following result. Result 3 (Baer [l]). A finite projective plane P is (P, 1)-transitive if and only if it is (P, I)-desarguesian. This result shows that the existence of axial collineations in a plane P is closely related to the geometric structure of the plane. It has as an immediate corollary the result that a finite projective plane is desarguesian if and only if it is (P, 1)-transitive for every point P and every line 1. But it also suggests many interesting questions. What, if anything, can one say if P is (P, 1)-transitive for some, but not all, choices of P and 1 ? What possibilities are there for P and 1 ? This latter question was answered by Lenz [9] and Barlotti [2] who gave a list of all possible configurations which may be formed by points P and lines 1 of a finite projective plane P for which P is (P. /)-transitive. This list, called the Lenz-Barlotti classification, provided one on the main areas of research in projective planes in t h e 196O's, when attempts were made to find examples of planes corresponding to each possible configuration listed in the classification. One particular possibility, prompted by the following result, has received special attention.

Result 4. Let P be a finite projective plane. If P is (A, 1)-transitive and (B, /)-transitive with A E 1, B E 1, A f B. then P is (C,1)-transitive for all C E 1. A line 1 for which a plane is (C, /)-transitive for all C E 1 is called a Iranslufion fine. Thus 1 is a translation line if and only if the group of elations with axis 1 is transitive on the points not on 1. A plane with a translation line is called a translation plane. A finite translation plane must have prime power order. If a finite plane has more than one translation line then, as a consequence of t h e Skornyakov-San Soucie Theorem, it is desarguesian. There are, however, many examples of non-desarguesian translation planes and a great deal is known about them. Anyone interested in projective planes in general should consult Hughes and Piper [ 6 ] ,but for more details on translation planes the reader should consult the excellent recent book of Luneburg [lo]. If n is any positive integer, n 3 2, and 9 is any prime power we will denote by P ( n , 9 ) the n-dimensional geometry over GF(q). (Thus P ( 2 , q ) is the desarguesian projective plane of order q.) A collineation a of P(n, q) is called axial if it fixes every point on a hyperplane and central if it fixes every hyperplane through a point. Result 1 is also true for collineations of projective

On axial automorphisms of symmetric desigm

335

geometries. For any point P and hyperplane x the group of collineations with centre P and axis x is transitive on the points of any line I through P (except of course for P itself and In x if PkZ x ) . Thus, using the phrase in an analogous way as for planes, any projective geometry is (P, xktransitive for all P and x. Furthermore the group of elations with axis x is transitive on the points not on x. In the study of projective geometries researchers have mainly been interested in their collineation groups and, once again, axial collineations have played a crucial role. The aim of this note is to make the (obvious) observation that projective planes and geometries are special examples of symmetric designs, and to state, (without giving any proofs), some of the recent results on axial automorphisms which attempt to generalise the results for planes and geometries to arbitrary symmetric designs.

2. Basic concepts A 2-(u, k , A ) design D is a finite incidence structure with a set of o points and b blocks such that (a) each block is incident with k 3 2 points and (b) any two distinct points are incident with A common blocks. The integers u, k, A are called the parameters of D. (These axioms imply that b = u(u - l ) A / k ( k - 1 ) and that each point is incident with r (=bk/u) blocks.) Fisher’s Inequality says b 2 u and a 2-(u, k, A ) design with b = u is called symmetric. In a symmetric design two distinct blocks cannot be incident with the same set of points so we may, if we wish, regard the blocks as point sets. Since b = u, u = ( k ( k - 1)+ A)/A and a symmetric design has only two independent parameters. If we put A = 1 and write n + 1 = k, then a symmetric design with A = 1 is a 2-(nZ+ n + 1, n + 1, 1) design. This is a finite projective plane of order n (with the lines as blocks). Another important family of symmetric designs are obtained from the projective geometries P(n, q). (This time we take the hyperplanes as blocks.) This gives us a symmetric

design for any prime power q 2 2 and any positive integer n 5 2. For the rest of this paper for any prime power q and integer n 3 2, P(n, q ) will represent this design. In view of the importance of these families it is not surprising that there are many results which give characterizations of them. Before stating one, we must make two more definitions. If P, Q are distinct points of a design D the line PQ is defined to be the intersection of all blocks containing P and 0.

3M

F. Piper

Similarly, if A, B. C are three points not on a line, the plane ABC is the intersection of all blocks containing them. (Thus in the symmetric design P(n, 4) lines and planes are exactly what they should be !) We can now state t h e Dembowski-Wagner characterization [4].

Result 5. If D is a 2-(u, k, A ) design such that some set of k points are not incident with a block then the following are equivalent: (i) D is a projective plane or is isomorphic to P(n, 4)for some n and q. (ii) Every line meets every block. (iii) Each line has (b - A ) / ( r - A ) points on it. (iv) Every plane is contained in the same number of blocks and D is symmetric. There are many o t h e r characterizations of these designs and we shall be considering some which are stated in terms of axial automorphisms. However, most of them merely show that their conditions imply one of (i), (ii), (iii) of (iv) in Result 5. An automorphism a of a symmetric design D is called axial if it fixes each point on a block (called its axis) and cenfrul if it fixes every block through a point (called its cenfre). Although any non-identity automorphism has at most one centre and at most one axis, it is no longer true that an automorphism has a centre if and only if it has an axis. This is one of the reasons why axial automorphisms of arbitrary symmetric designs have not proved so useful as for projective planes and geometries. It means, for instance, that we have three types of axial automorphism. An automorphism a # 1 with axis x is called a homology if it has a centre P with PfiZ 1, and elation if it has a centre P with P E 1 and a translation if it does not fix any point not on x . Thus a translation may or may not have a centre. (Note that elations are translations.) It is no longer true that t h e elations with axis x form a group. Result 6. I n a symmetric 2-design D t h e translations with a given axis x form a group.

Since no non-identity translation can fix a point which is not on the axis, the order of t h e group of translations with given axis x has order dividing u - k. We call x a franslation block if the group has order u - k, i.e., x is a translation block if the group of translations with axis x is transitive on the points not on x . Thus this is merely a generalization of the concept of a translation line in a projective plane. (Recall that, for a plane, lines and blocks are identical.) For any n and q every block of F(n, 4) is a translation block and furthermore, in these examples, t h e translations are all elations. If one assumes that all the translations are elations. then these are the only examples.

On axial automorphisms of symmetrk designs

337

Result 7. Let D be a symmetric 2-(u, k, A ) design with a translation block which is the axis of a group of elations of order u - k. Then, either A = 1 (in which case D is a translation plane) or D is isomorphic to P(n, q ) for some n and q. The above result is essentially due to Liineburg [ll]. In fact, Luneburg proved a much stronger theorem which has Result 7 as a corollary. For further details of Luneburg's results see [5] and for more basic information on designs see [5] or [7]. (But please do not hold your breath while waiting for the latter to appear !) We will now concentrate on translation blocks where the translations are not all elations.

3. Designs with translation blocks

Result 8 (Schulz [12]). If a symmetric 2-design D has a translation block, then it is a ( q n + l - 1 qn-'

q-1

qn-' - 1)

'q-1' q - 1

design for some prime power q and integer n

L 2.

Thus, Schulz showed that if a 2-design has a translation block, then it has the same parameters as one obtained from a projective geometry. In fact Schulz proved much more. He also showed that the translation group is a p-group and showed how to construct many examples. In view of Result 7, if A # 1 and the examples are not projective geometries, then some of the translations must have no centres. More recently Kelly [8] has studied symmetric designs with more than one translation block. He has shown that if every block is a translation block, then the design must be isomorphic to the points and hyperplanes of a projective geometry. He gives three constructions for designs with some, but definitely not all, translation blocks. (Each construction involves starting with a projective space and 'distorting' it.) He is then, rather surprisingly, able t o show that any symmetric design with more than one translation block can be obtained from a projective geometry by using a combination of his three constructions. Thus, in some sense, Kelly 'knows' all symmetric 2-designs with more than one translation block. His arguments are too detailed to include here and the reader is referred to [8].However, we will state two of his results which give further characterizations of P(n, q).

338

E Piper

Result 9. If a symmetric 2-design D has a set of translation blocks which meet in just one point. then it is isomorphic to the design obtained from the points and hyperplanes of a projective geometry, i.e., to P(n, 9)for some n and 9.

This is a natural generalization to arbitrary dimension of the Skornyakov-

San Soucie Theorem which says that a finite projective plane with two

translation lines is desarguesian. (Note that two lines must meet in a point !) It says that if D has a set of translation blocks which intersect in a point, then every block is a translation block. In fact, Kelly showed that, in general, the existence of even fewer translation blocks is sufficient to characterize the designs obtained from projective geometries. By Result 8 any symmetric design D with a translation block has parameters ?"+I-

9-1

1 q" - 1 '9-1'

p -1 9-1

)

for some prime power 9 and positive integer n 3 2. We will denote a design with these parameters by D(n, 9).

Result 10. If a symmetric design D ( n , q ) with q odd, n a 3 , has a set of translation blocks which intersect in a line, the D(n, q ) is isomorphic to P(n, 9). The case where 9 is even is rather different. However Kelly is able to determine exactly what happens if 9 is even and D(n, 9) is not isomorphic to P(n, 9). He is also able to give examples of 'new' designs which have translation blocks intersecting in point sets which are larger than lines. Thus, his Results 9 and 10 give best possible characterizations of P(n, q ) in terms of translation blocks.

4. Designs with axial automorphisms A symmetric 2-design D is (P,x)-transitive if, for any pair of points A, B with P # A, Pf B, A fiZ x, B fZ x and PA = PB, there exists an automorphism a with centre P and axis x such that A" = B. It is not t o o difficult to begin a

Lenz-Barlotti type classification for arbitrary symmetric designs. The problem is that in assuming that D is (P, x)-transitive one is assuming the existence of the maximum possible number of homologies or elations with centre P and axis .Y. Since any automorphism with centre P fixed every line through P this is essentially assuming something about the lengths of the lines through P. While there is nothing wrong with this, one is quickly led to the situation where the

On axial automorphisms of symmetric designs

339

centres and axes form subdesigns which, by using the Dembowski-Wagner Theorem, can be shown to be isomorphic to P(n, q ) for some n and q. It seems better to look for another concept which is equivalent to (P, /)-transitivity for projective planes but which allows for the existence of axial automorphisms with no centre. Butler [3] makes the following definition. If x and y are two distinct blocks of a symmetric design D, then D is (x fl y, x)-transitive if there is a group of automorphisms with axis x transitive on the points of y\x. If D is a projective plane and x and y are lines, then x fl y is a point P and (x fl y, x)-transitive means there is a group of (P,x)-elations transitive on the points of y other than P. This, clearly, is equivalent to (PIx)-transitive. Of course for symmetric designs other than planes the automorphisms with axis x need not have centres. This definition leads to a rather nice generalization of Result 4. Result 11. Let D be a symmetric 2-(v, k, A ) design. If D is (x f l y , x)-transitive and (x f l z, x)-transitive where x fl y # x fl z, then D is (x f l u, x)-transitive for all blocks u # x, i.e., x is a translation block.

Butler also generalizes Schulz's Theorem to get the following result. Result 12. Let D be a symmetric 2-(u, k,A ) design with A > 1. If D is (x n y, x> transitive for a pair of blocks x, y, then D has the parameters of P ( n , p ) for some n and q.

Given these two results he is able to produce the Lenz-Barlotti list for arbitrary symmetric designs. Now, just as for projective planes in the 1960's, the problem is trying to find examples of each type. Westfield College University of London London, England

References [l] R. Baer, Projectivities with fixed points on every line of the plane, Bull. Amer. Math. Soc. 52 (1946)273-286. [2] A. Barlotti, Le possibili configurazioni del sistema delle coppie punto-retta (A, a ) per cui un piano grafico risulta (A,ahtransitivo, Boll. Un. Mat. Ital. 12 (1957) 212-226. [3] N. Butler, Ph.D. "lesis, London University, 1981. [4] H.P. Dembowski and A. Wagner, Some characterizations of finite projective spaces, Arch. Math. 11 (1960)465-469. [5] H.P. Dembowski, Finite Geometries (Springer, Berlin, 1968).

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[6] D.R. Hughes and F.C. Piper, Projective Planes. Graduate Texts in Mathematics (Springer, Berlin, 1973). [7] D.R. Hughes and F.C. Piper, The theory of designs, to appear. [8]G. Kelly, Symmetric designs with translation blocks, to appear. [9] H. Lenz, Kleiner desarguesscher Satz und Dualitat in projektiven Ebenen, h e r . Dtsch. Math-Ver. 57 (1954) 20-31. 101 H. Liineburg, Translation Planes (Springer, Berlin, 1980). 111 H. Liineburg. Zentrale Automorphismen von A-Raumen, Arch. Math. 12 (1%1) 134-145. 121 R.H. Schulz, &r Blockplane mit transitiver Dilatationsgruppe, Math. Z. 98 (1%7) 60-82.

Annals of Discrete Mathematics 15 (1982) 341-355 @ North-Holland Publishing Company

PICTURES IN LATTICE THEORY Ivan RIVAL* and Bill SANDS Dedicated with affection to Nathan Mendelsohn on the occasion of his 65th birthday

We present a selection of results, both old and new, displaying the use and appeal of lattice diagrams.

1. Introduction

Pictorial aids to reasoning are generally ignored in scholarly mathematics. While geometrical diagrams may be tolerated in early mathematical training (as for example ‘Venn’ diagrams t o illustrate intersection patterns of sets) they are certainly suppressed in public by the time mathematical maturity sets in. After all, pictures may oversimplify and distort the real situation and mislead us from our mission. Still, it is a widespread practice, enjoyed by most practicing researchers in the privacy of the study, to use pictorial aids of one kind or another to assist in understanding and t o prompt discovery. However, when such discoveries appear in print the pictures are gone. Only exposition remains, obscuring what was, upon a time, simple and clear. Admittedly, pictures in print are not feasible in many branches of mathematics. A happy exception is lattice theory. In this paper, after a review of the concept of a lattice diagram, our aim is to tell a few illustrated tales, mostly familiar but some new, demonstrating the utility and appeal of pictures in lattice theory.

2. Graphs, digraphs and diagrams In a world without pictures, we might be presented with a lattice L as a set of ordered pairs; for example,

I. Rival. B. Sands

.w1

where u. h , . . . . h are the elements of L and ( a , b ) , for instance, means that a s h in L. Or, we might be presented this lattice L itemized in the form of an incidence matrix (Fig. A) where, for instance, the entry in the first row and second column i s I because a =sb, while the entry in the second row and first column i s 0 because h# a. Perhaps the most natural scheme for ‘drawing’ L would be as a graph whose vertices are t h e elements of L, and where two vertices are joined by an edge whenever the corresponding elements of L are comparable. The reflexive comparability graph of L is depicted in Fig. B. It is customary to suppress the -reflexivity’ of the order relation and thereby the loops at every vertex. The cymparubility graph of L is depicted in Fig. C. Unlike t h e incidence matrix of L, the comparability graph of L does not determine L. The simple device of ‘directing’ the edges of this graph will solve this problem. The directed comparubiliry graph of L is depicted in Fig. D. Actually. this graph contains more information than is really necessary t o recover L. We can exploit the ’transitivity’ of t h e order relation o n L to eliminate certain of t h e directed edges. For instance, since a < b and b < e we know that u < e without actually including the edge from a to e. Indeed, the

a

h

c

d

e

f

g

h

1

1

1 0

1 0

1

1

1

1

I I 0 1 O I 0 I 0 I 0 1 1 0 0 1 0 0 0 0 0 I 0 0 0 0 0 1 0 0 0 0 0

I I I 1 I I I

0 1 0 0 0 0 0 0 0 0 0 0 0 0

Fig. B.

Fig. A.

f Fig. C .

d Fig. D.

Pictures in lattice theory

343

only edges that we really need are those which correspond to pairs of elements where one ‘covers’ the other. In this way we construct the covering graph and the directed covering graph of L (Fig. E). A final artifice exploits the ‘antisymmetry’ of the order relation and makes it possible t o orient the directed covering graph in such a way that all arrows make an angle 8 with the horizontal satisfying 0” < 8 < 180”; this done we simply dispose of the arrows. The diagram of L ends up as depicted in Fig. F. It is evident that the actual pictorial representation of L in the plane can be carried out in many ways; for instance, Fig. G is a particularly unlikely alternative. What we obviously mean by the diagram of an ordered set is the equivalence class of all such pictorial representations. 0

e

C

Fig. E.

8

f

h

9 d

e C

6

d

0

Fig. F.

b

0

Fig.G.

The very first results in lattice theory pertain t o the two five-element lattices of Fig. 1. The results are these: “A lattice is modular if and only if it contains

I. Riwl. B. Sands

*5

M3 Fig. 1.

n o sublattice isomorphic to N5" 171; "A modular lattice is distributive if and only if it contains n o sublattice isomorphic to M," [3]. Up to isomorphism, each of the lattices Ns and M3 is determined by its covering graph: i n fact, each of these lattices is determined, up to isomorphism, wen by its comparability graph. This graph connection can be exploited in a number o f ways. For instance, it is a fact that requires some computation that "a modular lattice of finite length is distributive if and only if its covering graph contains n o subgraph (graph) isomorphic to t h e covering graph of M3" 1271. A second example concerns the lattice Sub L of a11 sublattices of a lattice L (ordered by set inclusion). "Let L and M be lattices such that Sub L = Sub M. Then L is modular (distributive) if and only if M is modular (distributive)" [lo]. Using the comparability graph t h e idea of the proof is particularly transparent [ 181. Observe that the vertices of the comparability graph of L are precisely the singleton sublattices of L, that is, the elements of height one in Sub L, and the edges of the comparability graph of L are determined by the ordering of t h e elements of height one and two in Sub L. Now, if L contains a sublattice {u. h. c. tf. e } = M3 then {u. h}. {a,c ) , { a . d } , {u, e } , (6,e } , {c. e ) , { d , e} are elements of height two in Sub L while the elements {b} v {c}, { b }v { d } , {c}v {d}have height greater than two in Sub L. If Sub L = Sub M then M, too, contains a sublattice isomorphic to M3. Similarly, if Sub L =Sub M, and L contains a sublattice isomorphic to N5.then M does too. For infinite lattices, there is a variant of MI that is particularly useful. Let M, denote the lattice of length two with countably many elements of height one (see Fig. 2). "Every infinite ordered set contains either an infinite chain or an infinite antichain." (This fact is easy to prove by some simple computation or by an appeal to Ramsey's theorem.) In t h e case that the ordered set is a lattice more

Pictures in lartice theory

I :j

345

1

I

I I t

2

I I

3

I4

2

I I

1 MW

w

I

I

wd

Fig. 2.

is true: “Every infinite lattice contains a sublattice isomorphic to w or ador Mu” [28],cf. [ll].This fact for lattices has several interesting consequences. There is a longstanding conjecture that any variety of finite height (in the lattice of varieties of lattices) is generated by a finite lattice. Now, an infinite lattice of finite length must contain a sublattice isomorphic to M , whence it contains the infinite chain of sublattices M3C M4C M C - * , each of which generates a distinct variety. It follows that “a variety of finite height is generated either by a finite lattice or by a lattice of infinite length”. Elsewhere [22] we conjecture that any infinite ordered set P of finite length must contain an infinite antichain A such that every x E P is comparable with either precisely none of the elements of A or precisely one element of A or infinitely many of the elements of A. In the case that P is a lattice we can choose A to be the infinitely many elements of height one in the sublattice of P isomorphic to iV&,. (Without the assumption that P is a lattice the conjecture is open.) 4. Herringbones

The lattice H illustrated in Fig. 3 is called the herringbone. Together with its variants it has been uncommonly well-used in recent years, as an example and as a counterexample. The most familiar variant of the herringbone is the three-generated lattice H3 illustrated in Fig. 4. Let us sample the usefulness of H3 itself. Why is the free lattice FL(3) on three unordered generators infinite ? Answer: because H3 is infinite. Why is M3 not projective ? Answer: because M3 is a homomorphic image of H3 but M3 is not a sublattice of H3.

1. Riuaf, 8. Sands

', I

I

" 3 Fig. 3.

Fig. 4.

Why is M, not transferable ? Answer: because M3 is a sublattice of the lattice of ideals of H$(the dual of H3) but M3 is not a sublattice of H $ [12]. Other herringbone variants provide quick counterexamples to some current conjectures. Davey and Rival [6] showed that every lattice generated by a three-element antichain contains one of finitely many, finite lattices, each generated by a three-element antichain, as a sublattice. A conjecture: For a finite ordered set P, every lattice generated by P contains a finite sublattice generated by P. Counterexample: The lattice illustrated in Fig. 5 is generated by a fourelement antichain, and every sublattice generated by a four-element antichain is infinite (Sands, unpublished). Another conjecture: For a finite ordered set P there is a finite list 2 ( P ) of finite lattices. each generated by P, such that every finite lattice generated by P confains a member of 2 ( P ) as a sublattice. Counterexample: Let P be the six-element antichain and, for any integer n 2 2, let If,,be the finite lattice illustrated in Fig. 6. Then H,, is generated by a six-element antichain, and every six-element antichain of H,, generates all of H,,. Hence, Y ( P ) does not exist (Sands, unpublished). Yet another 'herringbone-like' lattice provides a startling proof of this important result: "There are 2yo pairwise nonisomorphic three-generated lattices" [5].The lattice that accounts for this fact is illustrated in Fig. 7. To every subset S of the natural numbers we associate the three-generated lattice obtained by identifying the pairs {(ai,b,)l i E S } . In this way we obtain 2"O pairwise nonisomorphic homomorphic images of t h e lattice in Fig. 7 [23].

Pictures in lattice theory

Fig. 5.

347

Fig. 6 .

Fig. 7.

348

I. R i d B.Sands

The herringbone H is itself fairly ubiquitous: “Every finitely generated infinite lattice of finite width contains a subset isomorphic to H or Hd”[19]. In fact, every finitely generated infinite lattice of width three contains a subfam‘ce isomorphic to H or Hd. From this it follows that “every finitely generated, subdirectly irreducible lattice of width three is finite” [ 171.

5. Drawing free lattices

We have amply demonstrated already that a lattice need not be finite to be ‘drawable’. Beyond this, the question of when a lattice can be drawn is undoubtedly unanswerable. For example, it is hard to imagine what would be an effective diagram of a dense chain, say the rational numbers in the unit interval; on the other hand, it is feasible to ‘draw’ a picture of a lattice which contains a dense chain ‘down its spine’ (see Fig. 8) [24]. For which ordered sets P can the free lattice FL(P) generated by P be ‘drawn’ ? While this question is still less than precise, surprisingly it can be supplied

Fig. 8.

Pictures in lattice theory

Fig. 10.

349

I. Rival, B. Sands

350

with a rather satisfactory answer provided only that we concede that FL(3), and any lattice containing FL(3), cannot be drawn. There would likely be little resistance to this concession for FL(3) contains FL(N0) and FL(3) contains dense chains. Let n1+ n2+ * * * + nk denote the cardinal sum of k chains of sizes nl, n2,. . . , nkr respectively. Sorkin [26] had shown that FL(nl + nz) is finite if and only if nl nz =s 3 (cf. Fig. 9). In contrast Rolf [25] observed that FL(nl + nz) contains a sublattice isomorphic to FL(3) just if nl n 2 3 5. In the case that n1 nz = 4 Rolf [25] did provide an explicit representation of FL(nl + nz) (see Fig. 10). Rival and Wille [24] have shown that F y P ) can be drawn (by actually drawing it) if and only if P contains no subset isomorphic to 1+ 1+ 1 or 2 3 or 1+ 5. There is even a finite ordered set H such that FL(P) can be drawn if and only if FL(P) is isomorphic to a sublattice of F'L(H) (see Fig. 11).

-

-

+

H

Pichues in lanice theory

35 1

One consequence of such pictorial diversions is this: “For a finite ordered set P, the lattice variety generated by FL(P)is proper if and only if (PI > 1 and P contains no subset isomorphic to 1+ 1+ 1, 2 + 3, or 1+ 5” [24]. (A variety of lattices is proper if it contains 2 but does not contain all lattices.) 6. Planar lattices and dimension

A finite lattice is planar if it has a diagram in which none of the straight line segments intersect. The lattice 3 x 3 illustrated in Fig. 12 is planar. (Of course, not all diagrams of 3 X 3 avoid intersecting line segments.) A detailed analysis of the geometry of planar lattices and their planar embeddings led Kelly and Rival [14] to this characterization. Let 2’ be the family of lattices illustrated in Fig. 13, together with their duals. Then a finite

A non- planar embedding of 3 X 3.

A planar embedding of 3 X 3. Fig. 12.

Fig. 13.

352

I. Rival. B. Sands

lattice is planar if and only if it contains no subset isomorphic to a member of 2. The superficial impression that the theme of planar lattices is just a graph theoretical curiosity is a false one. The first clue that there is much more to this theme lies in the following twin observations: (i) a lattice is planar if and only if it has dimension at most two, (ii) each member of Y has dimension three. Recall. an ordered set P has dimension n ( d i m P = n ) if n is the least number of linear extensions of P whose intersection is P; P is irreducible if dim(P - { u } )< 12 for each a E P. For instance, the only irreducible ordered sets of dimension one and two are, respectively, the antichains 1 and 1 + 1. In contrast, the description of all irreducible ordered sets of dimension three seemed, until recently, quite intractable. In fact, the key idea to the complete description of all irreducible ordered sets of dimension three is the planarity of lattices. For an ordered set P, let N ( P ) denote the completion by cuts of P and, for a lattice L. let P ( L ) denote the subset of L consisting of all join irreducible or meet irreducible elements of L. There are several important observations regarding N ( P ) and P ( L ) . (iii) (Banaschewski [2]): “For every finite lattice L, N ( P ( L ) )= L.” (iv) (Baker, Fishburn. and Roberts [l]):“For every ordered set P, dim P = dim N ( P ) . ” (v) (Kelly and Rival [ 141): “An ordered set P has dimension at most two if and only if N ( P ) contains no subset isomorphic to a member of 3.’’ (vi) (Kelly and Rival [lS]): “For each member L of 2, P ( L ) (the shaded elements in Fig. 13) is an irreducible ordered set of dimension three.” Kelly [ 131 settled the whole matter by way of a detailed analysis which began with these observations (i) to (vi). Now, if P has dimension three then N ( P ) contains a subset isomorphic to P ( L ) for some L E 2. If P is, in addition, irreducible then P ( N ( P ) )= P. Furthermore, each element of P ( L ) is both the join of a subset of P and the meet of a subset of P. What remains is to reconstruct P by analysing the possible ways in which P ( L ) is a subset of N ( P ) . The final result is t h e following: An ordered set is irreducible of dimension three if and only if it is isomorphic to P ( L ) for some L E 2‘ or it is isomorphic to one of the ordered sets, or their duals, illustrated in Fig. 14. Finally, the unlikely combination of planarity and free lattices was exploited in t w o recent papers [20,21]. In the second of these papers it was shown that a finite sublattice of a free lattice is planar if and only if it contains no sublattice isomorphic to 2’ or Sn ( n 2 0) (see Fig. 15). It is now an almost immediate consequence of t h e diagram of Sn that any finite, subdirectly irreducible, breadth two. sublattice of a free lattice is planar.

Pictures in lanice rheory

353

\

Sn Fig. 15.

354

I. Rival. B. Sands

7. On the other hand.. . . It may be that one picture is worth more than ten thousand words. We end with three examples showing that, unfortunately, there are times when we need the ten thousand words! Is every uniquely complemented lattice Boolean ?‘ Draw any picture of a lattice in which every element has a unique complement and it will ‘surely’ be distributive. and so Boolean. Still, the situation in general is quite different as “every lattice can be embedded in a uniquely complemented lattice” [ 8 ] . In a geometric lattice of length n, is there an integer m =sn such that the number of elements of height m equals the width of the lattice ? As the smallest known counterexample has more than 60000 elements [9] it is rather unlikely that even a careful scrutiny of geometric lattice diagrams would settle the matter. Is the covering gruph of any finite lattice three-colourable ? Try it. The conjecture is somewhat off t h e mark; in fact, “for any positive integer n, there are finite lattices whose covering graphs are n-colourable” [4,161. Department of Mathematics and Statistics University of Calgary Calgary, Canada

References 111 K.A. Baker. P.C. Fishburn and F. Roberts. Partial orders of dimension 2. Networks 2 (1971) 11-28. [2] H. Banaschewski. Hiillensysteme und Erweiterungen v o n Quasi-Ordnungen. Z. Math. Logik Grundlag. Math. 2 (1956) 117-1M). [3) G. Birkhoff. On the lattice theory o f ideals, Bull. Amer. Math. SOC.40 (1934) 613-619. [4] B. Bollobb, Colouring lattices, Algebra Universalis 7 (1977) 313-314. IS] P. Crawley and R.A. Dean. Free lattices with infinite operations, Trans. Amer. Math. Soc. 92 (1959) 35-47. [6] B.A. Davey and I. Rival, Finite sublattices of three-generated lattices, J. Austral. Math. Soc. Ser. A 21 (1976) 171-178. 171 R. Dedekind. Lher die von drei Moduln erzeugte Dualgruppe, Math. Ann. 53 (1900) 371-403. 18) R.P. Dilworth. Lattices with unique complements, Trans. Amer. Math. Soc. 57 (1945) 123-154. 191 R.P. Dilworth and C. Greene. A counterexample to the generalization of Sperner’s theorem, J . Combin. Theory 10 (1971) l b 2 1 . [ 101 N.D. Filippov. Projectivity of lattices, Mat. Sb. 70(112) (1966) 3 6 5 4 ; English trans]., Amer. Math. Soc.Transl. 96 (2) (1970) 37-58. [ I I ] J Ginsburg and B. Sands, Minimal infinite topological spaces. Amer. Math. Monthly 86 (1979) 574-576. [ 121 G. Gratzer, Trends in Lattice Theory, J.C. Abbott, ed. (Van Nostrand-Reinhold, New York. 1970) pp. 173-215. [ 131 D. Kelly. The 3-irreducible partially ordered sets, Canad. J. Math. 29 (1977) 636-665.

Pictures in lattice theory

355

(141 D. Kelly and I. Rival. Planar lattices, Canad. J. Math. 27 (1975) 636-665.

(151 D. Kelly and I. Rival, Certain partially ordered sets of dimension three, J. Combin. Theory Ser. A 18 (1975) 239-242.

(161 J. NeBetiil and V. Rijdl, Combinatorial partitions of finite posets and lattices, Algebra Universalis, to appear. [17] W. Poguntke and B. Sands, O n finitely generated lattices of finite width, Canad. J. Math 33 (1981) M . [ 181 I. Rival, Projective images of modular (distributive, complemented) lattices are modular (distributive, complemented), Algebra Universalis 2 (1972) 395. [19] I. Rival, W. Ruckelshausen and B. Sands, On the ubiquity of herringbones in finitely generated lattices, Proc. h e r . Math. Soc. 82 (1981) 335-340. [20] I. Rival and B. Sands, Planar sublattices of a free lattice I, Canad. J. Math. 30 (1978) 1256- 1283. [21] I. Rival and B. Sands, Planar sublattices of a free lattice 11, Canad. J. Math. 31 (1979) 17-34. (221 I. Rival and B. Sands, O n the adjacency of vertices to the vertices of an infinite subgraph, J. London Math. SOC. 21(2) (1980) 393-400. [ U ] I. Rival and B. Sands, How many four-generated simple lattices ? Universal Algebra Semester, Stefan Banach Mathematical Centre Tracts, Warsaw, to appear. [24] I. Rival and R. Wille, Lattices freely generated by partially ordered sets: which can be ‘drawn’ ? J. Reine Angew. Math. 310 (1979) 56-80, [25] H. Rolf, The free lattice generated by a set of chains, Pacific J. Math. 8 (1958) 585-595. [26] Yu.Sorkin, Free unions of lattices, Mat. Sb. 30 (1952) 677-694. [27] M. Ward, The algebra of lattice functions, Duke Math. J. 5 (1939) 357-371. [28] T.P. Whaley, Large sublattices of a lattice, Pacific J. Math. 28 (1%9) 477-484. [29] R. Wille, On lattices freely generated by finite partially ordered sets, Coll. Math. SOC.J. Bolyai 17 (1977) 581-593.

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Annals of Discrete Mathematics 15 (1982) 357-369 @ North-Holland Publishing Company

ON MUTUALLY ORTHOGONAL RESOLUTIONS AND NEAR-RESOLUTIONS S.A. VANSTONE

Dedicated ro N.S. Mendelsohn on the occasion of his 65th birthday In this paper, we are interested in multiple resolutions and near-resolutions of balanced incomplete block designs. These resolutions must satisfy an orthogonality condition. Some of the known results are presented and some new families of mutually orthogonal nearresolutions of (0, k, k - ItBIBDs, including a complete listing of cyclic near-resolutions for all cyclic (16,3,2>BIBDs are given.

1. Introduction

A balanced incomplete block design (BIBD) D is a collection 9 3 of subsets (blocks) taken from a finite set V of elements (points) with the following properties: (a) Every pair of distinct elements from V is contained in precisely A blocks of 3. (b) Every block contzins exactly k elements. If 2) = I then we denote such a design as a (u,k, A)-BIBD. A (u, k, A)-BIBD D is said to be resolvable (RBIBD) if the blocks can be partitioned into classes R 1 ,R2,. . . , R, where r = (u - l)A/(k - 1) such that each point of D is contained in precisely one block of each class. The classes R , , RS,. . . , R, form a resolution of D. A (u, k, A)-BIBD is said to be doubly resolvable if there exist two resolutions R and R' of the blocks such that

v,

(Rin R]l s 1 for all Ri E R, R ] E R' (It should be noted that the blocks of the design are considered as being labelled so that if a subset of the point set occurs as a block more than once the blocks are treated as being distinct.) The resolutions R and R' are called orthogonal resolutions of the design. We denote such a design as a DR(u, k, A)BIBD. As an example, the array shown in Fig. 1 displays a pair of orthogonal 357

S.A. V-rone

358

-411

9 0 6

101 7 I

1

I

I

I

-

512

112 8

-

026

8 9 ll

3 3 4

6 13

131 5

7 810

122 3

I

0 4 5

9 10 12

1 3 7

Fig. I

resolutions of a (15,3,2>BIBD. The rows form one resolution and the columns form an orthogonal resolution. The existence question for DR(o, k. A>BIBDs is open. The following results are known.

Theorem 1.1. For all u = 0 (mod 2), u # 4 or 6, there exists a DR(u, 2, l)-BIBD. The designs of Theorem 1.1 are commonly referred to as Room squares. For an account of this result see [13].

Theorem 1.2. For p a prime power and n a positiue integer greater than 2, there exists a DR(p",p, 1)-BIBD. This result appears in [6]. Of particular interest is the case k = 3, A = 1. It is well known that for all u = 3 (mod 6), there exists a (u, 3, 1)-RF3IBD. The proof of this appears in [14]. The spectrum of DR(o, 3, 1)-BIBDs is not known. A necessary condition for existence is o = 3 (mod 6) but this is not sufficient. There is no DR(9,3, 1) or DR(15,3, 1). The existence of a DR(21,3, 1) is in doubt and some results on this case can be found in a paper by Mathon, Phelps

On mutually orthogonal resolutions and near-resolutions

359

and Rosa [ 111. Several non-isomorphic DR(27,3,l)s exist and can be found in [12]. The most general result, so far, appears in [6]. Theorem 1.3. For u = 3 or 9 (mod 24) and u sufficiently large, there exists a DR(u, 3, 1)-BIBD. In the case, k = 3, A = 2 more is known but the spectrum is still not entirely determined. This will be the subject of a later paper.

2. Mutually orthogonal resolutions and near-resolutions A set Q = {I?', R2,. . . , R'} of t resolutions of a (0, k, A)-BIBD is called a set of mutually orthogonal resolutions (MOR) if the resolutions of Q are pairwise orthogonal. For example, 3 MORs of an (8,2, 1)-BIBD are displayed in Fig. 3,.

4

-1

-2

-3

4

23 15

34

45

56

60

26

30

41

52

46

50

61

52

63

-5 01 63 04

=6 12 04 15

Fig. 2.

Two of the resolutions are obtained from the rows and columns of the array and t h e third is listed. Just as in the examples where we have used two orthogonal resolutions to coordinatize a two dimensional array, we could use t MORs to coordinatize a t dimensional hypercube. When t = 3, k = 2, A = 1, a set of MORs is commonly called a Room cube of side u - 1. The following result has recently been established by Dinitz and Stinson [4].

.MI

S.A. Vanstone

Theorem 2.1. For all u = 0 (mod 2), u f 4 or 6. there exists a Room cube of side 0-

1.

For t > 3, k

= 2.

A

=

1, Dinitz [5] has the following result.

Theorem 2.2. If q = 2k t + 1 is a prime power and t is odd, then there exists a set of t MORS of a (4+ 1.2. 1)-BIBD. The only upper bound known for t appears in [3].

Theorem 2.3. Let t be a set of MORs for a ( u , k . A)-BZBD. Then, t s A ( u - k ) / ( k -- 1). This bound is probably not sharp and we conjecture that t S i ( u - 2) when k=2andA=l. For t > 2, k > 2. A = 1. there is, at present, no value of u for which a set of t MORs of a (0,k , l)-BIBD exists. For t > 2, k > 2, A > 1, such sets do exist (see, for example. [ 101). A (u. k , A)-BIBD D is said to be near resolvable (NRBIBD) if the blocks of D can be partitioned into classes R 1 ,R 2 , .. . , R , such that for each point x of D there is precisely one class having n o block containing x and each class contains precisely u - 1 points of the design. For such a design to exist, a necessary condition is u = 1 (mod k ) and A = k - 1. In the case, k = 3, Hanani [S]has shown that this is also sufficient. Let R and R' be two resolutions of an NRBIBD. R and R' are again said to be orthogonal resolutions of the design provided

IR, n R;I s 1 for all Ri E R . R; E R'

If an NRBIBD has a pair of orthogonal resolutions, it is said to be doubly resolvable and is denoted DNR(0, k. A)-BIBD. These designs are very useful in recursive constructions for DR(u, k. A)-BIBDs and, hence, the existence question for them is of interest. As in the case o f doubly resolvable designs, we define a set of t mutually orthogonal near resolutions (MONR) to be r resolutions of an NRBIBD such that the resolutions are pairwise orthogonal. In the next section, we construct families of MONRs for various values of u. k and 1. 3. Starters and adders for near resolvable designs Let G = (0, g , , . . . , g,_,} be an additive abelian group of order rnt + 1. Let G* = G\{O}. A srarrer S of order m - 1 is a partition of G* into m-subsets,

O n mutually orthogonal resolutions and near-resolutions

361

( S , , S2,.. . , SI), such that these subsets are a set of base blocks for a cyclically generated BIBD with parameters (mt + 1, t(mt + l), mt, m, m - 1). If T is a subset of G* and a is an element of G*, then T + a is a set obtained by adding a to each element of T. An adder A(S) for the starter S is a set of elements ( a l ,a?,. . . , a,) of G*such that

u Si + ai = G* I

i=l

Theorem 3.1. If an abelian group G of order N = mt+ 1 admits a starter S of order m - 1 and an adder A(S), then there exists a DNR(v, m, m - 1)BIBD. The proof of this can be found in [9]. We also require other results from the same paper. For the sake of completeness, we give a proof of the first result.

Theorem 3.2. Let 9 = mt + 1 where 9 is a prime power, and let F = GF(9).Let T be the multiplicative subgroup of order t in F* = F\{O} and x be a primitive element of F. Ler M be an m-set whose elements form a system of distinct representatives for the cosets of T and whose differences are evenly distributed over the cosets of T. Then, S = (M, MX", MX'", . . . , MX('-l)"')is a starter. Furthermore, A(S)= (x", xm+", . . . , x ( t -

I)"+"

)

is an adder for S if and only if the elements of ( a + x": a E M ) lie in distinct cosets of T. h o o f . If A, B C allowed. Then,

E define A O B= (ab: a E A, b E B ) where repetitions are

since the elements of M form a system of distinct representatives for the cosets of T. Let B be the list of differences from M. Since the differences are evenly distributed, each coset is represented m - 1 times. Hence,

BXOU BX" U . *

*

U BX('-l)" = Bo

T = ( m - 1)F*.

Therefore, S = (M,MX",MX2"',. . . ,MX('-')"')is a starter for an (mt+ 1, m,m - 1)-BIBD. Now, suppose X o+ X",X' + X", . . . ,X("-')'+ X" are in distinct cosets of T. To show that A ( S ) is an adder for S, it is sufficient to show that for if j ,

-352

S.A. Vanstone

{i,j } C (0, 1. . . . , t - l}, MX'" a ~ i m +

+ Xi"+" and MXj" + Xfm+"are disjoint. Suppose

xi"+" = bXlm+ X'"+".

a, b E M

Then,

X'"'(u + X " ) = X'"(b

+X").

But. Xi".XI" E T and a + X", b + X " are in distinct cosets or else a = b. In either case, we have a contradiction. Therefore, A(S) is an adder for S. Conversely, suppose A(S) is an adder for S. Then, for a, b E M, a# b, Xi"(a + X " ) E MX'" + Xi"+" is different from b + X" E MXo+ X " for each i E {O, 1, . . . , t - 1). This implies that a + X " and b + X" are in distinct cosets of T. This completes the proof of the theorem. In GF(19). if we take M S

=

= {2O, 2', 2

) and n

=

13, then

(M2",M z 3 ,M26. M2', M2I2.M2")

and A(S) = ( ~ , 2 1 6 , 2 , 2 4 , 210) 2~,

yield a starter and adder for a D N R ( l 9 , 3 , 2 ) - B I B D .

Theorem 3.3. Let F = G F ( 9 ) where 9 is a prime power of the form mt + 1 with (m, t ) = 1 . Let M and T be the multiplicative subgroups of order m and t respectively in F* = F\{O}. Let x be a primitive element of F. Then, S

=

( M , Mx", Mx'"', . . . , Mx('-Ih 1

is a starter, Furthermore A(S) = (x", x"+", x Z m +)". . . x ( f -I ) m + n 1 is an adder for S if and only if x o + x", x ' + x", distinct cosets of T.

X"

+ x", . . . , x("-')' + x"

lie in

Theorem 3.4. Let 9 = 3t + 1 , (3, f ) = 1 and let F = GF(9). Let M and T be the subgroups of order 3 and t respectively. The cosets of M form a starter S(M, Mx3,Mx6, . . . , M x ~ ' - ~ ) . T = (xo, x 3 , x6, . . . , x ~ ' - ~a) permutation , of subgroup T is an adder if and only if2 is a cube in F, that is, 2 E T.

On mutually orthogonal resolutions and near-resolutions

363

These results are generalized to the case of t MONRs where t 3 3.

Theorem 3.5. Let S = (So, S1, . . . , Sf-l) be a starter of order m - 1 in an abelian group G of order mt+ 1. Let A@)= (&,al,.. . ,a,-l) and A'(S)= (ah, a ; , . . . , a;-l) be two adders for S. Then, there exist three MONRs of the (mt + 1, m, m - 1)-BIBDgenerated by S if the t elements ai - a:, 0 S i d t - 1 are distinct. Proof. Let R = { R i :i E G } where

where and where

U={u.:iEG}

U.= {S, + aj + i: 0 d j

d

t - l}, i E G ,

W = { W i :i E G } Wi = {Sj + a; + i :0 S j s t - 1) , i E G .

Since A(S) and A'(S) are adders, the resolution R is orthogonal to each of the resolutions U and W. Now, a starter S' for the resolution U is (So+ 6, Si+al,..., Sf-l+uI-l). Let A(S')=(ab-ao, a ; - u l ,..., aLl-a,-l). Since Wo (0 is the identity element of G ) is a resolution class of W, and the entries of A(S')are all distinct, it follows that A(S') is an adder for S' and the resolutions U and W are orthogonal. We call A(S) and A'(S)orthogonal adders. Lemma 3.6. Let q = mt + 1 be a prime power and (m, t ) = 1. Suppose xo+ x", x' x", xZf+ x", . . . ,x("-')' + x" are in distinct cosets of the multiplicative subgroup T, of order t, in F* = GF(q)\{O}. IThen xo+ F a ,x' + x"+' ,..., X(m-W + x"+' are in distinct cosets of T for 0 < i d m.

+

Suppose M and T are as defined in Theorem 3.3. If xo+x", x'+x",

. . . , X ( " ' - ~ ) ~ + X "are in distinct cosets, then Lemma 3.6 gives at least m

distinct adders for the starter S = (M, Mx", . . . ,M X ( ' - ' ~ We ) . now establish that this gives m + 1 MONRs.

S.A. Vanslone

364

Theorem 3.7. Let 9 = mt + 1 be a prime power and M and T be multiplicative subgroups of F* of orders m and t respectively. For the starter S = ( M , Mx", Mx'". . . . . MX"-)m ), ti)(^) = (xn+if, X m + n + i r , . . . . x ( f - 1)m t n t i f ) is an adder for each i , O S i s rn - I iff x"+ xn, x m + xn, , . . ,X ( ' - I ) ~ +are ~ in distinct cosets of T. Furthermore, for if j, (i, j } (0, 1,2, . . . , m - l}, the adders Ati)(S), A(I)(S) give orthogonal resolutions. Proof. From Theorem 3.3 and Lemma 3.6, A i ( S ) is an adder for each i, 0 s i s m - 1 . To show that A ( i ) ( S ) ,A ( j ) ( S )give orthogonal resolutions for i # j , we apply Theorem 3.5. Consider

suppose

1= X h m + n + r f(1 - x ( J - ' ) ' ) . Since i# j ,

1 - x ( J -' ) I #0 and, hence, 1 = h. Therefore, all differences must be distinct and A(')(S),A("(S) give XJm+n+if(

1 - -r(f-i)f

orthogonal resolutions.

Theorem 3.8. Let 9 = 3t + 1 be a prime power and (3, t ) = 1. If 2 is a cube in GF(9), then there exists a set of four MONRs for a cyclically generated (q, 3 . 2 > 8 1 8 0 ouer GF(9). Proof. The result follows from Theorems 3.4 and 3.7

Theorem 3.9. Let 9 = mt + 1 be a prime power and M and T be multiplicative subgroups of F* of orders m and t respectively. Suppose x o + x", X" + x", . . . , x ( ' - ' ) ~+ x" are in distinct cosets and xo+ XI. X" + x', . . . ,x('+~)"'+ X I are also in distinct cosets where 0 =Sn < 1 t - 1. Then A ( S ) = (x"+", x ~ + " + ~. '., . , X ( f - l ) m + n + i f

for each i, 0

)

i s m - 1 and

for each j , 0 d j zs m - 1 give orthogonal resolutions of the design generated from the starter S = ( M , Mx", . . . , Mx(f-I)m 1.

Proof. Consider Xhm*n+ir

- Xhmtl+/f =

Xhm+if(Xn

1.

- x/+(/-~)f

Since -mr < f + ( j - i)r < mt, and n # I, then x" - xf+('-')' # 0, and, so, by Theorem 3.5, A ( S ) and A ' ( S ) give orthogonal resolutions.

On mutuallv orthoxonal resolutions and neat-resoluh'ons

365

4. Some results on MONRs

Let A = {Rl, RZ,. . . ,R,} be a set of t MONRs for a (v, k, k - 1)BIBD. The set A is called a frame if the resolution classes in each Ri, 1d i < t, can be labelled R f , R:, . . . ,R ; so that

In other words, if the t resolutions are used to coordinatize a t-dimensional array, the main diagonal remains empty. Frames are very useful in recursive constructions. The MONRs of Section 3 are easily seen to be frames. It can be shown that infinite families of MONRs can be constructed by starting with a frame and applying a singular direct product construction or a PBD construction. These constructions, and the spectrum of these designs will be the subject of a later paper. Theorem 4.1. Suppose GF(q),q = mt + 1 admits a starter S of order m - 1 and a pair of orthogonal adders A(S), A'(S). Then GF(q"), n a positive integer, admits a starter of order m - 1 and a pair of orthogonal adders.

Proof. Let M be the multiplicative subgroup of order m in GF(q)\{O}. Since GF(q) is a subfield of GF(q"), M is a subgroup of GF(q")\{O}= F*. Let C = {co,cl, . . . , CI},1 = (4" - l)/m be a system of distinct representatives for the cosets of M in F*. Now, Sco U Sc, U . . . U ScI is a starter of order m - I in GF(q"). It is easily checked that

and

A(S)co U A(S)cl U . * . U A(S)CI A'(S)co U A'(S)cl U . * * U A'(S)CI

are orthogonal adders for this starter. The primes q such that q = 3t + 1, (3, t ) = 1, q < 500 and 2 is a cube in GF(9) are 31. 43, 157, 223, 229, 277, 283, 439, 457, and 499. Hence, Theorem 3.8 guarantees at least 4 MONR (q, 3,2)-BIBDs for each value of q in the above list. Theorem 3.8 does not apply to q = 37 with m = 3 since t = 12 and (3, 12) # 1. If we apply Theorem 3.1 to M = {1,2, 19) and take 2 as a primitive element, then S = (M,Mz3,M2', . . . , M233)

366

S.A. Vansrone

is a starter and

AJS) = (2n, 23+n,. . . ,

is an adder for n = 11, 27, 34 and 35. Therefore, there exists a DNR(37,3,2)BIBD with S MONRs. Theorem 3.9 can also be used t o construct 7 MONRs for the (31,3,2>BIBD generated from the starter consisting of the multiplicative subgroup of order 3 and its cosets. In this case we take n = 0 and I = 3. Using these techniques we can construct 5 MONRs for a (29,4,3)-BIBD over GF(29). 5. Cyclic MONRs for smali values of t~

It is easily seen that there is no set of t 3 2 MONRs for a (7,3,2>BIBD. M.J. and C.J. Colburn [2] have listed all (13,3,2>BIBDs that are cyclically generated. C.J. Colburn [2] has shown that there exists a cyclically generated DNR-(13.3,2).

S

=

((0, 1.3}, { 12,2, 8}, (9, 11,4}, ( 5 6 , 10)) and A(S)= (0, 10,6, 1) .

A complete computer search of all 10 cyclic (13,3,2)-BIBDs for cyclic nearresolutions shows that 7 are near resolvable and only one of these 7 had 2 orthogonal near-resolutions. Unfortunately, for the purposes of recursive

constructions, it does not produce a frame. M.J. and C.J. Colburn [2] list 89 cyclic (16,3,2>BIBDs. A computer check of these systems established that 72 of these systems have cyclic near-resolutions, 22 of them have cyclic orthogonal near-resolutions and only one admit more than 2 cyclic orthogonal near-resolutions. The following is a list of the 22 systems having at least 2 cyclic resolutions. Starter Adder

45 6

5

9 11 1.5 10 13 2 8 2

14 1 7 7

8 12 3 10

(2)

8910 5

1 3 7 1215 4 2 8

11 14 5 13

2 613 4

(3)

5 6 7 8

81015 1 1 1 4 2 11 9

1412 5

9133 15

(4)

56 7 3

9 11 1 1215 2 4 2

10 14 3 8

4 813

(5)

45 6 6

911 1 1215 3 12 5

71014 8

813 2

15 1

On mutually orthogonal solutions and near-resolurions

45 6 6

911 3 1215 8 14 110 13 2 7 8 1 5 12

56 8 2

2 310 10

1214 9 5

1115 4 7

13 1 7 8

12 3 8

1113 6 5

71015 11

912 5 7

4 814 1

12 3 5

1113 7 2

5 814 13

912 4 8

61015 4

56 7 5

1315 9 8

811 2 7

1 412 2

1014 3 10

23 5 7

8 913 9

1214 6 15

710 1 1115 4 13 4

56 8 4

1213 2 1

15 1 9 6

4 7 1 1 1014 3 13 8

34 6 92

7 814 3

911 15 1013 2 10 4

1 5 12 13

56 8 2

3 410 8

1113 1 1215 7 7 4

14 2 9 11

7 8 10 0

3 411 9

1214 1 5

5 915 6

2 613 12

56 8 2

910 2 11

1214 1 11 15 4 0 7

3 713 12

56 8 1

3 412 15

1315 9 0

34 6 2

1314 7 13

91115 4

2 510 7

812 1 6

7 8 11 1415 5 7 12

1 3 6 3

1012 2 11

913 4 15

23 6 1

1011 1 4

5 713 5

1214 9 13

4 815 9

34 7 4

5 615 15

91114 1

1012 2 7

13 1 8 5

7 1 1 1 14 2 9 13 3

367

.m

S.A. Vanstone

The final system is the only one which has more than 2 cyclic orthogonal near-resolutions. This system has precisely three such resolutions. (22)

2 3 6 1011 1 6 3 1 7

5 715 1214 9 12 9 6

0

4 813 2 2

All but 2 of the first 21 designs listed above are frames. The design number (22) has 3 cyclic orthogonal near resolutions but unfortunately no pair of these is a frame. Finally, we briefly consider cyclic MONRs of a (19,3,2)-BIBD. We will not give an exhaustive account. In an earlier example, we displayed the starter

and the adder A ( S ) = (3, 14,5, 17, 16,2).

From Theorem 3.9, with n = 6 , 8 , 13 and 15, we can p r o L x e a set a IONRs for this cyclic (19,3,2)-BIBD. A computer check of this system established that 5 is the maximum number of cyclic orthogonal near-resolutions. This system is a frame. 6. Conchdon

In Section 3 we constructed infinite classes of t MONRs for t > 2. It remains an open question whether there exist t MONRs of a (u, k, 1)BIBD for t > 2 , k >2. A good bound on the size of f remains an open problem. Note added in prod

Theorem 1.3 has recently been improved to “ u = 3 (mod 6) and u sufficiently large”. Also, a set of 3 MORs of a (255,3,1)-BIBD has been constructed. Department of Combinatorics and Optimization University of Waterloo Waterloo, Canada

On mutually orthogonal resolutions and near-resolutions

369

References [ 11 C. Colburn, Private communication, 1980. (21 M.J. Colburn and C.J. Colburn, Cyclic block designs with block size 3, European J. Combin., to appear. [3] M. Deza, R.C. Mullin and S.A. Vanstone, Orthogonal systems, Aequationes Math. 17(2/3) (ISnS) 322-330. [4] J. Dinitz and D.R. Stinson, The spectrum of Room cubes, European J. Combin., submitted. [5] J. Dinitz, New lower bounds for the number of painvise orthogonal symmetric latin squares, Congressus Numerantium XXII 1 (1979) 39S398. [6] R. Fuji-Hara and S.A. Vanstone, On the spectrum of doubly resolvable designs, Congressus Numerantium, 28 (1980) 3 W 7 . [7] K.B. Gross, R.C. Mullin and W.D. Wallis, The number of painvise orthogonal symmetric latin squares, Utilitas Math. 4 (1973) 23!9-251. [8] H. Hanani, On resolvable balanced incomplete block designs, J. Comb. Theory 17 (1974) 275289. [9] F. Hoffman, P.J. Schellenherg and S.A. Vanstone, A starter-adder approach to equidistant permutation arrays and generalized Room squares, Ars Combin. 1 (1976) 307-319. [lo] E. Kramer, S. Maglivcras and D. Mesner, Some resolutions of S(5,8,24), preprint, 1979. [ l l ] R. Mathon, K. Phelps and A. Rosa, A class of Steiner triple systems of order 21 and associated Kirkman systems, preprint, 1980. [I21 R. Mathon and S.A. Vanstone, On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays, Discrete Math. 30 (1980) 157-172. [13] R.C. Mullin and W.D. Wallis. The existence of Room squares, Aequationes Math. 13 (1975) 1-7. [14] D.K. Ray-Chauduri and R.M. Wilson, Solution of Kirkman’s school girl problem, Proc. Symposia in Pure Mathematics, h e r . Math. Soc. 19 (1971) 187-204.

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Annals of Discrete Mathematics 15 (1982) 371-376 @ North-Holland Publishing Company

APPENDIX. RESEARCH PAPERS BY N.S. MENDELSOW

[11 A group theoretic characterization of the general projective collineation group, Proc. Nat. Acad. Sci. 30 (1944)279-283;(MR 41,6(2) (1945)p. 41). [2]A psychological game, Amer. Math. Monthly 53(2) (1946). [3] Symbolic solution of card matching problems, Bull. h e r . Math. Soc. 52(10) (1946)918-924;(MR365,8(7) (1947)p. 365). [4] A group theoretic characterization of the general projective collineation group, Trans. Roy. SOC.Canada (3)40 (1946);(MR 526,8(9)(1947)p. 526). [5]Application of combinatorial formulae to generalization of Wilson’s theorem, Canad. J. Math. l(4) (1949)328-336;(MR 159,11(3)(1950)p. 159). [6]An application of a famous inequality, Amer. Math. Monthly 58(8) (1951). [7]Representations of positive real numbers by infinite sequence of integers, Trans. Roy. Soc. Canada (3)46 (1952)45-55;(MR 544,14(6) (1953)p. 544). [8] A problem in combinatorial analysis, Trans. Roy. SOC.Canada (3)47(1953) 21-26;(MR 387, 15(5) (1954)p. 387). [9]Some elementary properties of ill conditioned matrices and linear equations, Amer. Math. Monthly 63 (1956)285-295;(MR 1138,17(10) (1956)p. 1138). [lo] The asymptotic series for a certain class of permutation problems, Canad. J. Math. 8 (1956)234-244;(MR 935,17(9) (1956)p. 935). [ll] Non-desarguesian geometries satisfyingthe harmonic point axiom, Canad. J. Math. 8 (1956)532-562;(MR 665, 18(8) (1957)p. 665). [12] Some properties of approximate inverse of matrices, Trans. Roy. SOC. Canada (3)50 (1956)53-59;(MR 634,18(8) (1957)p. 634). [13] An iterative method for the solution of linear equations, M.T.A.C. ll(58) (1957); (MR 175,19(2) (1958)p. 175). [14]The computation of complex proper values and vectors of a real matrix with application to polynomials, M.T.A.C. ll(58) (1957);(MR 686,19(6)(1958)p.

686).

[ 151 Same generalizations of the problem of distinct representatives, Canad. J. Math. 10 (1958)230-241 (with A.L. Dulmage). [16] The convex hull of sub-permutation matrices, Proc. h e r . Math. SOC.9(2)

(1958)253-254.

[17] Coverings of bipartite graphs, Canad. J. Math. 10(1958)517-534 (with A.L. Dulmage).

37 1

377,

Research papers by

N.S.Mendelsohn

[18) The term and stochastic ranks of a matrix, Canad. J. Math. 11 (1959) 269-279; (MR 3433, 21(6) (1960) p. 638) (with A.L. Dulmage). [ l 9 ] A structure theory of bipartite graphs of finite exterior dimension, Trans. Roy. SOC.Canada (3) 53 (1059) 1-13 (with A.L. Dulmage). [2(4 A note on the stochastic rank of a bipartite graph, Canad. Math. Bull. 2(3) (1059) 159-162; (MR 7215, 21(11) (1960) p. 1332) (with A.L. Dulmage). [ ? I ] Orthogonal latin squares. Canad. Math. Bull. 2(3) (1959) 211-216 (with D. Johnson and A.L. Dulmage). [22] On an algorithm of Birkhoff concerning doubly stochastic matrices, Canad. Math. Bull. 3(3) (1960) 237-242; (MR A133,24A(IA) (1062) p. 23) (with D. Johnson and A.L. Dulmage). [23] Orthomorphisms of groups and orthogonal latin squares, Canad. J. Math. 13 (1961) 356-372; (MR A1544, 23A(3A) (1%2) p. 280) (with D. Johnson and A.L. Dulmage). [24] Permutations with confined displacements, Canad. Math. Bull. 4 (1961) 3-37. [2S] Those Stirling numbers again, Canad. Math. Bull. 4(2)(1961) 149-152; (MR A2336, 23A(4A) (1962) p. 443). 1261 Connectivity and reducibility of graphs, Canad. J. Math. 14 (1962) 529-539; (MR 3856, 25(5) (1963) p. 750) (with D. Johnson and A.L. Dulmage). 1271 Matrices associated with the Hitchcock problem, J. Assoc. Comput. Mach. 9(4) (1962) 409-418; (MR 6625,27(6) (1964) p. 1261) (with A.L. Dulmage). [28] On the inversion of sparse matrices, Math. Comp. (1962); (MR 6375, 27(6) (1964) p. 1219) (with A.L. Dulmage). [29] The exponent of a primitive matrix, Canad. Math. Bull. 3 3 ) (1%2) 241-244: (MR 135, 26(1) (1963) p. 28). [30]A remark on a result of Marvin Marcus, Canad. Math. Bull. 6(1) (1%3); (MR 3721, 26(4) (1963) p. 714) (with A.L. Dulmage). [31] Congruence relationships for integral recurrences, Canad. Math. Bull. 5(3) (1962); (MR 132, 26(1) (1%3) p. 28). [32] Two algorithms for bipartite graphs, SIAM 11 (1%3) 183-194; (MR 4224, 27(4) (1964) p. 811) (with A.L. Dulmage). [33] Remarkson solutions of the optimal assignment problem, SIAM 1l(4) (1963) 1103-1 109; (MR 2914, 30(3) (1%5) p. 551) (with A.L. Dulmage). [MI The characteristic equation of an irnprimitive matrix, SIAM l l ( 4 ) (1%3) 1034-1045; (MR 116, 29(1) (1965) pp. 23-24) (with A.L. Duimage). [35] Gaps in the exponent set of primitive matrices, Illinois J. Math. 8(4) (1964) 642-656; (MR 5872, 31(6) (1966) p. 1055) (with A.L. Dulmage). [36] The exponents of incidence matrices, Duke Math. J. 31(4) (1964) 575-584; (MR 3109, 30(4) (1965) p. 591) (with A.L. Dulmage). [37] An algorithmic solution for a word problem in group theory, Canad. J. Math. 16 (1964) 509-516; (MR 1248, 29(2) (1965) p. 245).

Research papers by N.S. MendeLrohn

373

[38] Some graphical properties of matrices with non-negative entries, Aequationes Math. 2 (1969) 150-162; (MR 7135,40(6) (1970) p. 1293) (with A.L. Dulmage). [39] The structure of powers of non-negative matrices, Canad. J. Math. 17 (1%5) 31&%330; (MR 4248, 35(4) (1968) p. 774) (with A.L. Dulmage). [40]A calculus for a certain class of word problems in groups, J. Combin. Theory l(2) (1966) 202-208; (MR 1380, 34(2) (1%7) p. 241) (with C.T. Benson). [41] A systematic method for combinatorial counts, 4th Symp. Appl. Math., Stud. Appl. Math. 4 (1967) 105-111. [42] Graphs and matrices, in: F. Harary, ed., Graph Theory and Theoretical Physics (Academic Press, New York, 1967) pp. 167-227; (MR 5468, 40(5) (1970) pp. 1003-1004). [43] A n application of matrix theory to a problem in universal algebra, Linear Algebra Appl. l(4) (1%8) 471-478; (MR 120, 39(1) (1970) pp. 22-23). [44]A combinatorial method for embedding a group in a semigroup, Proc. Symp. Pure Math. (1968) 157-165; (MR 159, 48(1) (1974) p. 33). [45] Hamiltonian decomposition of the complete directed n-graph, in: Theory of Graphs, Proc. Colloq. Tihany, Hungary (1966) pp. 237-241; (MR 4361, 38(5) (1%9) p. 776). [46] A natural generalization of Steiner triple systems, in: Computers in Number Theory (Academic Press, New York, 1971) pp. 323-338; (MR 122, 48(1) (1974) p. 27). [47] Combinatorial designs as models of universal algebra, in: Recent Progress in Combinatorics (Academic Press, New York, 1969) pp. 123-132; (MR 85, 41(1) (1971) p. 17). [48] Some examples of man-machines interaction in the solution of mathematical problems, in: Computational Problems in Abstract Algebra (Pergamon Press, Oxford, 1970) pp. 217-223; (MR 2713, 42(2) (1971) p. 487). [49] Defining relations for subgroups of finite index of groups with a finite presentation, in: Computational Problems in Abstract Algebra (Pergamon Press, Oxford, 1WO) pp. 43-45; (MR 3575, 41(3) (1971) p. 659). [50] Benson Mendelsohn algorithm for certain word problems in groups, IBM Proc., 1968. [51] Free subgroups of groups with a single defining relation, Arch. Math. 19 (1%9) 577-580; (MR 4258, 39(4) (1970) p. 773) (with Rimhak Ree). [52] Directed graphs with the unique path property, Combinatorial Theory and its Applications I1 (1%9) pp. 783-799; (MR 6690, 45(5) (1973) p. 1233). [53] Planarity properties of the Good-deBruijn graphs, Proc. Calgary Internat. Conf. Combinatorics and their Applications (1%9) 177-185; (MR 1711, 42(2) (1971) p. 305) (with Diane Johnson).

Research papers by N.S.Mendelsohn

374

[541 Latin squares orthogonal to their transposes, J. Combin. Theory 2(2) (1971) 187-189; (MR 88, 45(1) (1973) p. 17). [55] Intersection numbers of t-designs, in: Mirsky, ed., Studies in Pure Mathematics (Academic Press, New York, 1971) pp. 145-150; (MR 5819, 45(5) (1971) pp. 1058-1059). [56] Orthogonal Steiner systems, Aequationes Math. 5 (1970) 268-272; (MR 1587, 44(2) (1972) p. 301). [S7] A theorem on Steiner systems, Canad. J. Math. 22(5) (1970) 1010-1015; (MR 1677, 42(2) (1971) p. 299). (581 A linear diophantine equation with applications to non-negative matrices, Ann. New York Acad. Sci. 175 (1970) 287-294; (MR 221, 42(1) (1971) p.

40).

[59] A single groupoid identity for Steiner loops, Aequationes Math. 6 (1971) 228-230; (MR 6969, 45(5) (1973) p. 1279). [ a ] Applications of intersection numbers to 1-designs, Proc. 2nd Chapel Hill Conference o n Combinatorial Mathematics and its Applications, 1970; (MR 4994, 45(4) (1973) p. 913). I611 Some results on ordered quadruple systems, Proc. Louisiana Conf. on Combinatorics, Graph Theory and Computing, (1970) 297-309; (MR 5827, 42(51) (1971) p. 1060) (with R.G. Stanton). [62] A single identity for boolean ,groups and boolean rings, J. Algebra 20 (1972) 77-82; (MR 2689, 44(3) (1972) p. 506) (with R. Padmanabhan). [63] On maximal sets of mutually orthogonal idempotent latin squares, Canad. Math. Bull. 14(3) (1971) (MR 8865, 46(6) (1973) p. 1533). [MJ Extended triple systems, Aequationes Math. 8(3) (1972) 291-299; (MR 3211, 47(3) (1974) p. 561) (with D.M. Johnson). [65] On the Steiner systems S(3,4, 14) and S(4,5, 15), Utilitas Math. 1 (1972) 5-95; (MR 1618, 46(2) (1973) p. 271) (with S. Hung). [66]Construction of perpendicular Steiner quasigroups, Aequationes Math. 9(2/3) (1973) 150-156; (MR 6305, 48(4) (1974) p. 1102) (with C.C. Lindner). [67] Directed triple systems, J. Combin. Theory 14(3) (1973) 310-318; (MR 3190, 47(3) (1974) p. 557). [a]Inequalities for t-designs with repeated blocks, Aequationes Math. 10(2/3) (1974) 212-222; (MR 12539, 55(6) (1978) p. 1693). [69] Handcuffed designs, Aequationes Math. 2(2/3) (1974) 256-266; (MR 188, 51(1) (1976) p. 27) (with S. Hung). [70] The golden ratio and van der Waerden's theorem, Proc. 5th South-East Conf. on Combinatorics, Graph Theory and Computing (1974) 93-109; (MR 5545, 51(3) (1976) p. 780). [71] On Howell designs, J. of Comb. Theory (A) 16 (1974) 174-198; (MR 8271, 48(5) (1974) p. 1440) (with S. Hung).

Research papers by N.S. Mendelsohn

375

[72] Construction of cyclic quasigroups and applications, Aequationes Math. 14 (1976) 111-121 (with C.C. Lindner). [73] Minimal identities for boolean groups, J. Algebra 34 (1975) 451-457; (MR 2798, 53(2) (1977) p. 393) (with R. Padmanabhan). [74] Groupoid varieties such that every 2-generated groupoid in the variety has fixed finite order, unpublished. [75] Commutators in free groups, unpublished. [76] The equation d(x)= k, Math. Mag. 49 (1976) 37-39; (MR 252, 53(1) (1977) p. 34). [77] Algebraic construction of combinatorial designs, Proc. U. of T. Seminar on Algebraic Aspects of Combinatorics; (MR 5439, 52(3) (1976) p. 773). 178) A polynomial map preserving the finite basis property, J. Algebra 48 (1977); (MR 3045, 57(2) (1979) p. 393) (with R. Padmanabhan). [79] Handcuffed designs, Discrete Math. 18 (1977) 23-33; (MR 5318, 56(3) (1978) p. 726) (with S. Hung). [80] Perfect cyclic designs, Discrete Math. 20 (1977) 63-68. [81] The spectrum of idempotent varieties of algebras with binary operators based on two variable identities, Aequationes Math. 18 (1978) 330-332. [82] On the existence of extended triple systems, Utilitas Math. 14 (1978) 249-267 (with F.E. Bennett). 1831 On pure cyclic triple systems and semi-symmetric quasigroups, Ars Combin. 5 (1978) 15-22 (with F.E. Bennett). [84] Self-orthogonal Weisner designs, 2nd Internat. Conf. on Combinatorial Mathematics, Ann. New York Acad. Sci. 319 (1979) 391-3%. [85] Some remarks on 2-designs S2(2,3, v ) , Proc. 9th South-East Conf. on Combinatorics, Graph Theory and Computing (1978) 119-127 (with F.E. Bennett). [86]On the construction of Schroeder quasigroups (with C.C. Lindner and S.R. Sun) to appear. [87] Identities preserved by group divisible designs (with C.C. Lindner) to be submitted. [88]Construction of totally symmetric quasigroups with a specified number of idempotents, Utilitas Math. 15 (1979) 33-50 (with F.E. Bennett). [89] A class of combinatorial quasigroups, Math. Rep. Acad. Sci. 1 (1979) 13-17. [90]Resolvable perfect cyclic designs, J. Combin. Theory Ser. A 29(2) (1980) 142-150 (with F.E. Bennett and E. Mendelsohn). [91] Orthogonal latin square graphs, J. Graph Theory 3 (1979) 325-338 (with C.C. Lindner, E. Mendelsohn and B. Wolk). [92] On the spectrum of Stein quasigroups, Bull. Austral. Math. Soc. (with F.E. Bennett) to appear.

376

Reseanh papers b y N.S.Mendelsohn

1931 Conjugate orthogonal latin square graphs, Proc. 9th South-East Conf. on Combinatorics Graph Theory and Computing (1979) 179-192 (with F.E. Bennett). [94] Division problem in the Stein quasi-variety (1980) (with R. Padrnanabhan) t o appear.

Annals of Discrete Mathematics Previous Volumes in this Series

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Studies in Integer Programming edited by P.L. HAMMER, E.L. JOHNSON, B.H. KORTE and G.L. NEMHAUSER 1977 viii + 562 pages

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Algorithmic Aspects of Combinatorics edited by B.ALSPACH, P. HELL and D.J. MILLER 1978 out of print

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Advances in Graph Theory edited by B. BOLLOBAS 1978 viii + 295 pages

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Discrete Optimization, Part I edited by P.L. HAMMER, E.J. JOHNSON and B. KORTE 1979 xii + 299 pages

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Combinatorial Mathematics, Optimal Designs and their Applications edited by J. SRIVASTAVA 1980 viii + 391 pages

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Topics on Steiner Systems edited by C.C. LINDNER and A. ROSA 1980 x + 3 4 9 pages

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Combinatorics 79, Part I edited by M. DEZA and I.G. ROSENBERG 1980 xxii + 309 pages

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Combinatorics 79, Part I1 edited by M. DEZA and I.G. ROSENBERG 1980 viii + 309 pages

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