(2,k)-Factor-Critical Graphs and Toughness

Graphs and Combinatorics (1999) 15 : 137±142

Graphs and Combinatorics ( Springer-Verlag 1999

…2; k†-Factor-Critical Graphs and Toughness Mao-Cheng Cai1*, Odile Favaron2, and Hao Li2 1 Institute 2 LRI,

of Systems Science, Academia Sinica, Beijing 100080, China Bat. 490, Universite Paris-Sud, 91405 Orsay cedex, France

Abstract. A graph is …r; k†-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. with an r-regular spanning subgraph). We show that every ttough graph of order n with t V 2 is …2; k†-factor-critical for every non-negative integer k U minf2t ÿ 2; n ÿ 3g, thus proving a conjecture as well as generalizing the main result of Liu and Yu in [4].

1. Introduction All graphs G ˆ …V ; E† under consideration are simple and ®nite of order jV j ˆ n. If A J V , we denote by G ÿ A the subgraph obtained from G by deleting the vertices in A together with the edges incident with vertices in A. For a vertex x of G; NA …x† is the set of neighbors of x in A and dA …x† ˆ jNA …x†j. If the two subsets A and B of V are disjoint, E…A; B† is the set of edges between A and B and e…A; B† ˆ jE…A; B†j. We denote by d…G† and k…G† the minimum degree and the vertex connectivity of G. If S is a cutset of G, o…G ÿ S† is the number of connected components of G ÿ S. When G is not complete, its toughness is de®ned by  jSj S is a cutset of G . It is clear from the de®nition that t…G† :ˆ min o…G ÿ S†

d…G† V k…G† V 2t…G†. The graph G is said to be t-tough for every positive t U t…G†. For a clique Kn , we usually put k…Kn † ˆ n ÿ 1 and t…Kn † ˆ y. An rfactor of G, where r is a positive integer, is an r-regular spanning subgraph of G. In particular, a 1-factor is a perfect matching. When r is odd, only graphs of even order can admit an r-factor. Tutte gave necessary and su½cient conditions for a graph to have an r-factor ([6] for r ˆ 1 and [7] for r V 2). Let us recall these conditions for r V 2. For a given positive integer r and a pair S; T of disjoint subsets of V, a component C of G ÿ …S U T† is said to be odd if rjV …C†j ‡ e…T; V …C†† is odd. We denote by o1 …S; T† the number of odd components of G ÿ …S U T† and let * Research partially supported by National Natural Science Foundation of China and by the CoopeÂration Franco-Chinoise PRA 93-M10

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qG …S; T† :ˆ rjSj ÿ rjTj ‡

X

dGÿS …v† ÿ o1 …S; T†:

vAT

Note that when r is even, the odd components C are de®ned by e…T; V …C†† odd and, inP particular, e…T; V …C†† V 1 for an odd component C. This implies that for r dGÿS …v† ÿ o1 …S; T† V 0 and thus qG …S; T† V rjSj ÿ rjTj. even, vAT

Theorem A. (Tutte [7]): (i) qG …S; T† has always the same parity as nr. (ii) G has an r-factor if and only if qG …S; T† V 0 for every pair S; T of disjoint subsets of V. The graph G is said …r; k†-factor-critical if G ÿ X admits an r-factor for every subset X of k elements of V (when r ˆ 1, we usually simply say k-factor-critical). For r V 2, the study of these graphs has been initialized by Liu and Yu [4] under the name of …r; k†-extendable graphs. We prefer to keep here the term factorcritical because usually, in the term extendable, X is not any subset of k vertices but must satisfy some given properties. Using Tutte's Theorem, Liu and Yu found the following characterization of …r; k†-factor-critical graphs. Theorem B. (Liu and Yu [4]): Let r; k be integers with r V 2 and k V 0, and G a graph of order n V r ‡ k ‡ 1. Then G is …r; k†-factor-critical if and only if qG …S; T† V rk for any pair S; T of disjoint subsets of V with jSj V k. They also gave a su½cient condition for a graph to be …2; k†-factor-critical in terms of its toughness. Theorem C. (Liu and Yu [4]): Let G be a graph of order n and toughness t…G† V 3. Then G is …2; k†-factor-critical for every integer k such that 3 U k U t…G† and k U n ÿ 3. However they think Theorem C is not best possible and propose Conjecture D. (Liu and Yu [4]): Let G be a graph of order n and toughness t…G†. If t…G† V q and n V 2q ‡ 1 for some integer q V 1, then G is …2; 2q ÿ 2†factor-critical. Our purpose is to prove this conjecture with the necessary restriction t…G† V 2 since it is proved in [1] that for any positive real number e there exists an …r ÿ e†tough graph which has no r-factor. 2. The main result Theorem 1. Let G be a t-tough graph of order n with t V 2. Then G is …2; k†-factorcritical for every non-negative integer k U minf2t ÿ 2; n ÿ 3g and the bound 2t ÿ 2 on k is sharp. Proof: First we may assume G to be not complete for otherwise the result is obvious. Now suppose, to the contrary, that G is not …2; k†-factor-critical. Then, by

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Theorem B, there exist a pair S; T of disjoint subsets of V with jSj V k such that qG …S; T† < 2k, implying by Theorem A(i) qG …S; T† U 2k ÿ 2: Among all such pairs, we choose a pair S; T with additional properties: (i) qG …S; T† is minimum and (ii) subject to (i), T is minimal under inclusion. P dGÿS …v† ˆ 0; o1 …S; q† ˆ 0, and thus qG …S; T† ˆ Then T 0 q. For otherwise vAT

2jSj V 2k, a contradiction. We write U :ˆ V …G† ÿ …S U T†;

jSj ˆ s;

jTj ˆ t:

Thus s V k and t V 1. In order to prove the theorem we need the following claims, whose ideas already appeared in [4]. Claim 1: (a) T is independent. (b) For all y A T, each edge of E…y; U† joins y to an odd component of G ÿ …S U T†, and, moreover, di¨erent edges of E… y; U† join y to di¨erent odd components of G ÿ …S U T†. Proof of Claim 1: For any y A T, let L… y† be the component of G ÿ …S U …T n fyg†† containing y, and put h…y† ˆ 1 if L… y† is odd for the pair S; T n fyg, 0 otherwise. Let also u…y† be the number of odd components of GP ÿ …S U T† joined to y by some edges. Then qG …S; T n fyg† ˆ 2s ÿ 2…t ÿ 1† ‡ dGÿS …v† ÿ …o1 …S; T† ÿ u… y† ‡ h… y††. On the other hand, qG …S; T n fyg† V v A Tnfyg

qG …S; T† ‡ 2 by the choice of …S; T† and by parity. Hence 2 UqG …S; T n fyg† ÿ qG …S; T† ˆ 2 ÿ dGÿS … y† ‡ u…y† ÿ h…y† ˆ 2 ÿ dT …y† ÿ e…y; U† ‡ u… y† ÿ h…y†. But dT … y† V 0, e…y; U† ÿ u…y† V 0 and h…y† V 0. Therefore, dT … y† ˆ 0, implying (a), h…y† ˆ 0 and e…y; U† ˆ u…y†, implying (b). Note that by (b), there is no edge between T and the even components of G ÿ …S U T†. Claim 2: For t V 2, s ‡ e…T; U† ÿ o1 …S; T† V tt. Proof of Claim 2: Let us label the odd components Ci of G ÿ …S U T† in such a way that for i U o2 , each component Ci contains at least one vertex with exactly one neighbor in T, and for i > o2 , no vertex of Ci has exactly one neighbor in T. For short we put o1 …S; T† ˆ o1 and note that 0 U o2 U o1 . For each i U o2 , we choose in Ci one vertex ui such that dT …ui † ˆ 1 and put L ˆ fu1 ; u2 ;


; uo2 g. If o2 ˆ 0 then L ˆ q. We denote by W the set NU …T† (by Claim 1(b), W J 6 V …Ci †) and put Z ˆ W nL. Each vertex of W has at least one 1UiUo1

neighbor in T. Moreover by the de®nition of o2 and the fact that e…T; Ci † is odd for all i, each component Ci with i > o2 contains at least one vertex having at least three neighbors in T. Therefore e…T; U† V jW j ‡ 2…o1 ÿ o2 † ˆ jZj ‡ o1 ‡

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…o1 ÿ o2 † V jZj ‡ o1 . The equality e…T; U† ˆ jZj ‡ o1 occurs if and only if o2 ˆ o1 and each vertex of W has exactly one neighbor in T. On the other hand, by the construction of Z, the t vertices of the set T, which is independent by Claim 1, belong to t di¨erent components of G ÿ …S U Z†. Since t V 2, S U Z is a cutset of G and thus jS U Zj V tt. Hence s ‡ e…T; U† ÿ o1 …S; T† V tt, as required. Claim 3: For t V 2 and s ˆ 2; s ‡ e…T; U† ÿ o1 …S; T† > tt. Proof of Claim 3: Suppose s ‡ e…T; U† ÿ o1 …S; T† ˆ tt. Then by the proof of Claim 2, o2 ˆ o1 , every vertex of W ˆ NU …T† has exactly one neighbor in T, and tt ˆ jS U Zj, that is, the number of components of G ÿ …S U Z† is exactly t (in particular G ÿ …S U T† has no even component). Now, since d…G† V 2t…G† V 4 and s ˆ 2, and by Claim 1(b), some vertex y of T has one neighbor in at least two di¨erent components Ci , say y1 in C1 and y2 in C2 . In the choice of L, let us take u1 ˆ y1 and u2 ˆ y2 . For the set R ˆ S U Z U fyg of cardinality jZj ‡ 3, the number of components of G ÿ R is at least …t ÿ 1† ‡ 2 ˆ t ‡ 1, and thus t…t ‡ 1† U jZj ‡ 3. This contradicts tt ˆ jS U Zj ˆ jZj ‡ 2 since t V 2. We distinguish three cases according to the values of t. Case t ˆ 1. Then s V 2t ÿ 1. Indeed, say T ˆ f yg. If e…y; U† U 1, clearly s V 2t ÿ 1. And if e…y; U† V 2, then, by Claim 1(b), o1 …S; T† ˆ e… y; U† and S U fyg is a cutset, implying s V to1 …S; T† ÿ 1 V 2t ÿ 1. Hence s V k ‡ 1 and qG …S; T† V 2s ÿ 2t V 2k, a contradiction. Case t ˆ 2. By Claim 1, either E…T; U† is empty or there exists an odd number of edges, and thus exactly one edge, between T and each odd component of G ÿ …S U T†. Therefore the two vertices of T belong to two di¨erent components of G ÿ S and S is a cutset. Hence s V 2t V k ‡ 2 and thus qG …S; T† V 2s ÿ 2t V 2k, a contradiction, Case t V 3. Then qG …S; T† ˆ s ÿ 2t ‡ …s ‡ e…T; U† ÿ o1 …S; T†† V s ÿ 2t ‡ tt by Claim 2, with a strict inequality when s ˆ 2 by Claim 3. As t V 2, then k U s U qG …S; T† U 2k ÿ 2

…†

implying k V 2. Let us show k V 3. Indeed, if k ˆ 2, the equality occurs everywhere in …† and s ˆ 2, contradicting Claim 3. k‡2 , we have As s V k; t V 3 and t V 2   k‡2 5k ÿ2 ˆ ÿ 3 > 2k ÿ 2; qG …S; T† V k ‡ 3 2 2 a contradiction. Since the assumption qG …S; T† < 2k leads to a contradiction in all cases, qG …S; T† V 2k and thus G is …2; k†-factor-critical by Theorem B. The proof is complete. To show the sharpness of our result, consider the graph G consisting of a clique of vertex set A ˆ S U fyg with jSj ˆ k V 3, a second clique C of order at least 2,

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all the edges between C and S, and one edge yz for some vertex z A C. The graph k‡1 V 2 and is not …2; k†-factor-critical since G ÿ S has G has toughness t…G† ˆ 2 no 2-factor. Hence a t-tough graph with t V 2 and n V 2t ‡ 2 is not necessarily …2; 2t ÿ 1†-factor-critical. r An obvious consequence of Theorem 1 is the following. Corollary 3. (conjecture D): Let q be an integer V 1 and G a graph of order n V 2q ‡ 1 and toughness t…G† V maxfq; 2g. Then G is …2; 2q ÿ 2†-factor-critical.  Note that  for the values of t…G† belonging to intervals of the form 2q ‡ 1 ; q ‡ 1 , Theorem 1 shows that G is …2; 2q ÿ 1†-factor-critical and thus is 2 slightly stronger than Conjecture D. 3. Open problem For r ˆ 1, a theorem similar to Theorem 1 already exists: Theorem E. (Favaron [3]). Let G be a t-tough graph of order n with t > 1. Then G is …1; k†-factor-critical for every non-negative integer k such that n ‡ k is even, k < 2t and k U n ÿ 2. It would be interesting to generalize Theorems E and 1 to larger values of r and to determine functions t0 … f † and k0 …t; f † such that any t-tough graph with t > t0 … f † is …r; k†-factor-critical for every non-negative integer k with …n ‡ k† f even, k U k0 …t; f † and k U n ÿ … f ‡ 1†. Added, as a partial answer to the open problem, the following results have been obtained [5, 8]: (1) Every t-tough graph of order n V 12 with t V 4 is …3; k†-factor-critical for every non-negative integer k such that n ‡ k even and k U minf2t ÿ 3; n ÿ 7g. (2) Every t-tough graph of order n V 14 with t V 5 is …4; k†-factor-critical for every non-negative integer k U minf2t ÿ 4; n ÿ 8g. Acknowledgments. This work was done while the ®rst author was visiting LRI, Universite Paris-Sud, he wishes to thank LRI and Professor Hao Li for their hospitality. The authors are grateful to Professor H. Enomoto for his stimulating discussions, which led to Theorem 1 extended to case t ˆ 2.

References 1. Enomoto, H., Jackson, B., Katerinis P., and Saito, A.: Toughness and the existence of k-factors, J. Graph Theory 9, 87±95 (1985) 2. Enomoto, H.: Toughness and the existence of k-factors III, Discrete Math. 189, 277± 282 (1998)

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3. Favaron, O.: On k-factor-critical graphs, Discussiones Mathematicae-Graph Theory 16, 41±51, (1996) 4. Liu G., Yu, Q.: k-factors and extendability with prescribed components, submitted 5. Shi, M., Yuan, X., and Cai, M.: …3; k†-Factor-critical graphs and toughness, submitted 6. Tutte, W.T.: The factorization of linear graphs, J. London Math. Soc. 22, 107±111, (1947) 7. Tutte, W.T.: The factors of graphs, Canad. J. Math. 4, 314±328, (1952) 8. Yuan, X., Shi, M., and Cai, M.: …4; k†-Factor-critical graphs and toughness, in preparation

Received: December 16, 1996 Revised: September 17, 1997