4-Quasiperiodic Functions on Graphs and Hypergraphs

Cybernetics and Systems Analysis, Vol. 37, No. 3, 2001

BRIEF COMMUNICATIONS 4-QUASIPERIODIC FUNCTIONS ON GRAPHS AND HYPERGRAPHS

O. G. Rudenskaya

UDC 419.1

Sets of solutions of the equation f ( x ) − f ( y) = k in N 2 are described, where f is a function strictly monotone and 4-quasiperiodic in N . The descriptions obtained are used in investigating the genus of a complete bipartite graph. Keywords: problems of discrete mathematics, 4-quasiperiodic function, genus of complete bipartite graphs, representation of parameters of graphs.

In many problems of discrete mathematics (minimization of disjunctive normal forms, coding, covering problem, etc.), it is necessary to determine the genus of a complete bipartite graph K m , n for a fixed m or n [1], the thickness of a graph K n , n [2], the thickness of an n-dimensional unit cube [3], the degree of each node in a clique of rank 5, the chromatic number of a partial hypergraph generated by the set of nodes of the clique of rank 5 [4], and the greatest cardinality of matching in hypergraphs of some classes [4]. All these quantities are functions having a general property. Let us formulate them. A function f ( x ) defined on a set D is called 4-quasiperiodic in D if (∃T )(∀x ∈ D):[ x + 4 ∈ D ⇒ f ( x + 4) = f ( x ) + T ] . When D = N 0 , such functions are described by the following theorem. THEOREM. For any odd q, the set of all 4-quasiperiodic functions defined in N 0 can be given by the formula

f ( n) = a 0 n + a1 ( − 1)

n

+ a2 (

qn    2 − 1)  

+ a3 (

 q n +1    2  − 1) 

+ a4 ,

(1)

where a 0 , a1 , . . . , a 4 ∈R and T = 4a 0 . When q is odd, any 4-quasiperiodic function f ( n) defined in N 0 is representable in the form (1), where the coefficients of q = 1(mod 4) are defined as a 0 = [ f ( 4) − f (0)] / 4 , a1 = [ f ( 4) − 2 f (3) + 2 f (2) − 2 f (1) + f (0)] / 8 ,

(2)

a 2 = [ f ( 4) − f (3) − f (2) + f (1)] / 4 , a 3 = [ f (3) − f (2) − f (1) + f (0)] / 4,

(3)

a 4 = − [3 f ( 4) − 2 f (3) − 2 f (2) − 2 f (1) − 5 f (0)] / 8; when q ≡ 3(mod 4), only the following coefficients are changed: a 2 = [ f (3) − f (2) − f (1) + f (0)] / 4 , a 3 = [ f ( 4) − f (3) − f (2) + f (1)] / 4. In the particular case where T = 0, we have f ( 4) = f (0).

Cybernetics Institute, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 169-172, May-June, 2001. Original article submitted April 5, 1999. 1060-0396/01/3703-0443$25.00

©

2001 Plenum Publishing Corporation

443

TABLE 1 Group Points that are solutions of Eq. (6) for number points (5) ∅

1 2

(1, 1)

j

i 1-3

1-4

1-3

1-4

Points from N 2 that form the set M when τ = h + p and p = 0,1,... ∅ The points of the straight line y = x − 4h

3

(i + j, j)

1-3

1-4

(i + j + 4τ, j + 4 p)

4

(i + j, j), (i + j + 1, j + 1)

1-3

1-3

(i + j + 4τ, j + 4 p), (i + j + 1 + 4τ, j + 4 p)

5

(i + j, j), (i + j + 1, j + 1), (i + j + 2, j + 2)

1,3

1,2

(i + j + 4τ, j + 4 p), (i + j + 1 + 4τ, j + 4 p), (i + j + 2 + 4τ, j + 2 + 4 p)

6

(i + j, j), (i + j + 1, j + 1), (i + j + 3, j + 2)

1,2

7

(i + j, j), (i + j + 1, j + 2)

2,3

1,2

8

(i + j, j), (i + j + 2, j + 1)

1,2

1-3

(i + j + 4τ, j + 4 p), (i + j + 2 + 4τ, j + 1 + 4 p)

9

(i + j, j), (i + j + 2, j + 1), (i + j + 3, j + 2)

1,2

1,2

(i + j + 4τ, j + 4 p), (i + j + 2 + 4τ, j + 1 + 4 p), (i + j + 3 + 4τ, j + 2 + 4 p)

1,2

(i + j + 4τ, j + 4 p), (i + j + 1 + 4τ, j + 4 p), (i + j + 3 + 4τ, j + 2 + 4 p) (i + j + 4τ, j + 4 p), (i + j + 1 + 4τ, j + 2 + 4 p)

10

(i + j, j), (i + j + 2, j + 2)

1-3

1,2

(i + j + 4τ, j + 4 p), (i + j + 2 + 4τ, j + 2 + 4 p)

11

(i + j, j), (i + j + 3, j + 2)

1,2

1,2

(i + j + 4τ, j + 4 p), (i + j + 3 + 4τ, j + 2 + 4 p)

12

(i + 1, 1), (i + 2, 2), (i + 3, 3), (i + 4, 4)

1-3

1-4

The points of the straight line y = x − 4h − i

13

(i + 1, 1), (i + 2, 2), (i + 4, 4)

1,3

1-4

(i + 1 + 4τ, j + 4 p), (i + 2 + 4τ, 2 + 4 p), (i + 4 + 4τ, 4 + 4 p)

14

(i + 1, 1), (i + 3, 2), (i + 4, 4)

1,2

1-4

(i + 1 + 4τ, 1 + 4 p), (i + 3 + 4τ, 2 + 4 p), (i + 4 + 4τ, 4 + 4 p)

15

(i + 1, 1), (i + 3, 3), (i + 4, 4)

1,3

1-4

(i + 1 + 4τ, 1 + 4 p), (i + 3 + 4τ, 3 + 4 p), (i + 4 + 4τ, 4 + 4 p)

16

(i + 1, 1), (i + 4, 4)

1-3

1-4

(i + 1 + 4τ, 1 + 4 p), (i + 4 + 4τ, 4 + 4 p)

17

(i + 1, i), (i + 4, i + 1)

1-3

1-4

(i + 1 + 4τ, i + 4 p), (i + 4 + 4τ, i + 1 + 4 p)

18

(i + 2, 1), (i + 3, 2), (i + 4, 4)

1,2

1-4

(i + 2 + 4τ, 1 + 4 p), (i + 3 + 4τ, 2 + 4 p), (i + 4 + 4τ, 4 + 4 p)

19

(i + 2, 1), (i + 3, 3), (i + 4, 4)

1,2

1-4

(i + 2 + 4τ, 1 + 4 p), (i + 3 + 4τ, 3 + 4 p), (i + 4 + 4τ, 4 + 4 p)

20

(i + 2, 1), (i + 3, 3), (i + 5, 4)

1,2

1-4

(i + 2 + 4τ, 1 + 4 p), (i + 3 + 4τ, 3 + 4 p), (i + 5 + 4τ, 4 + 4 p)

21

(i + 2, 1),(i + 4, 4)

1,2

1-4

(i + 2 + 4τ, 1 + 4 p), (i + 4 + 4τ, 4 + 4 p)

22

(4,1), (5,4)

1-3

1-4

( 4 + 4τ, 1 + 4 p), (5 + 4τ, 4 + 4 p)

p It is obvious that f ( n + 4) = f ( n) + 4a 0 and T = 4a 0 for function (1). However, if the values of f ( n), n = 0 , 1, 2 , 3, 4, are known for this 4-quasiperiodic function, then, for each odd q, the coefficients of representation (1) are found from the system obtained after sequential substitution of n = 0 , 1, 2 , 3, 4 in (1). p Let us solve the equation (4) f ( x ) − f ( y) = k in N 2 , where k ≥ 0 and f is an increasing function that is 4-quasiperiodic in N 0 . When f is a decreasing function, we multiply both sides of the equation by – 1 and pass to the equivalent equation with the increasing function − f . When k < 0, we obtain solutions of Eq. (4) in the form of points symmetric about the solutions of the equation f ( x ) − f ( y) = − k with respect to the straight line y = x . When k = 0, all the points from N 2 that belong to the straight line y = x and only these points are solutions of Eq. (4). k Let k > 0. Denote by h the greatest solution of the inequality f (1 + 4 z) − f (1) ≤ k in N. It is obvious that h =   . T  Using the definition of a 4-quasiperiodic function f and taking into account its increasing character, we compile Table 1 of description of the set M of all solutions of Eq. (4) depending on whether the points M 0 (1, 1) and M 4(i −1) + j (i + j , j), 444

(5)

Fig. 1 TABLE 2 k = ∆f

∆n

k  (when h =   ) T 

k ≡ 0( mod T ) k ≡ f (i + j) − f ( j) ( mod T ), i = 1, 2, 3; j = 1, 2, 3, 4

n0

(when p = 0,1, 2,...)

4h

1 + 4p

4h + i

j + 4p

TABLE 3 k = ∆ν

∆n

k  (when h =   ) 5 

n0

(when p = 0,1, 2,...)

k ≡ 0( mod 5)

4h

k ≡1( mod 5)

4h + 1

1 + 4 p or 3 + 4 p, or 4 + 4 p

4h + 1

2 + 4p

k ≡ 2( mod 5)

p

4h + 2

3 + 4 p or 4 + 4 p

k ≡ 3( mod 5)

4h + 2

1 + 4 p or 2 + 4 p

4h + 3

3 + 4p

k ≡ 4( mod 5)

4h + 3

1 + 4 p or 2 + 4 p, or 4 + 4 p

where i =1, 2 , 3 and j =1, 2 , 3, 4 (see Fig. 1), are solutions of the comparison f ( x ) − f ( y) = k − Th.

(6)

We reduce 79 probable cases to 22 groups (see Table 1). According to (5) and (6), Eq. (4) is solvable only for the values of k that are presented in Table 2. Considering k as an increment ∆f of the function f ( n), we specify the values of the increment ∆n (in Table 2) that corresponds to k and the values of n 0 that assume them. In using this table, we take into account that values of k can coincide for different points (i + j , j). Example. Let us find all possible positive values of the increment ∆ν of the genus of a complete bipartite graph K 7, n , the corresponding values of the increment ∆n, and the values of n 0 .  5( n − 2)  It is shown in [1] that ν( n) = ν( K 7, n ) =   . The function ν( n) is increasing and 4-quasiperiodic (with T = 5)  4  in N 0 . We can easily obtain Table 3 from Table 1 or Table 2. Thus, many parameters of various classes of graphs can be represented in the form of dynamics of increments of functions that have a rather general type defined as a 4-quasiperiodic function.

445

REFERENCES 1. 2. 3. 4.

446

G. Ringel, “Das Geschlecht des vollstandigen paaren Graphen,” in: Abh. Math. Sem. Univ., 28, Hamburg (1965), pp. 139-150. L. W. Beineke, “Complete bipartite graphs: decomposition into planar subgraphs,” in: Seminar in Graph Theory, F. Harary (ed.), New York (1967), pp. 42-53. M. Kleinert, “Die Dicke des n-dimensionalen Wuerfel-Graphen,” in: Graph Theory: Coverings, Packings, and Tournaments [Russian translation], Mir, Moscow (1974), pp. 145-150. C. Berge, Graphes et Hypergraphes, Dunod, Paris (1970).