On a bound for the maximum number of C8s in a 4–cycle free bipartite

On a bound for the maximum number of C80 s in a 4–cycle free bipartite graph

GENE FIORINI and FELIX LAZEBNIK Department of Mathematical Sciences University of Delaware Newark, DE 19716 Abstract. Let G be a 4–cycle free bipartite graph on 2n vertices with partitions of equal cardinality n having e edges. Let c8 (G) denote the number of cycles of length 
n e  f ( n ) , where f (t) = t(t−1)(t− 8 in G. We prove that for n ≥ 4, c8 (G) ≤ 3 n4 − 4! 2)(4n−3−3t). If G is extremal with respect to the number of 8–cycles, then rn −2 < √  n
n  4n−3 e 1 − ≤ r , where r = + . This implies that c (G) ≤ 3 f (r − 2) . n n 8 n n 2 2 4! 4 Furthermore, if Gq is the incidence point–line graph of a finite projective plane of

2 
n  order q, and nq = q 2 + q + 1, then c8 (Gq ) = n2q 2q = 3 n4q − 4!q f (rnq ) , and Gq is “close” to being extremal in this sense. Section 1: Introduction. Let G = Gn denote a family of simple graphs of order n. For a simple graph H and G ∈ G, let (G, H) denote the number of subgraphs of G isomorphic to H. Let h(n) = h(G, H, n) = max{(G, H)|G ∈ G} and G(H, n) = {G ∈ G|(G, H) = h(n)}. We will refer to graphs of G(H, n) as extremal. The problem of finding h(G, H, n) and G(H, n), for fixed G, H, n, has been studied extensively and is considered as central in extremal graph theory. Though it is hopeless in whole generality, some of its instances have been solved. Often the results are concerned with bounds on hn and partial description of the extremal graphs. For example, if Km denotes the 1

complete graph of order m, H = K2 , and G is the family of all graphs of order n which contain no Km as a subgraph, 3 ≤ m ≤ n, then the solution is given by the famous Tur´ an Theorem. For the same H, if Ks,t denotes the complete bipartite graph with partition class sizes s, t and G is the family of all (m, n)–bipartite graphs with no Ks,t , we have the, so called, Zarankiewicz problem. These and many other examples can be found in [2]. For some later results see [4,5,6]. All missing definitions can be found in [2]. Let V (G) and E(G) denote the set of vertices and edges of a graph G, e = e(G) = |E(G)|. The neighborhood of a vertex v ∈ V (G) is denoted by N (v) (v 6∈ N (v)), and the degree of vertex v in G by degG (v). S For S ⊆ V (G), define N (S) by N (S) = N (v). If G contains a cycle, the girth of v∈S

G is the length of a shortest cycle in G. For a positive integer n, n ≥ 2, let G6 (n, n) be the class of bipartite graph on 2n vertices with partitions of equal cardinality n and girth at least 6 ( i.e. 4–cycle free). Let G ∈ G6 (n, n) have partition (V1 (G), V2 (G)) such that V1 (G) = {u1 , . . . , un }, V2 (G) = {v1 , . . . , vn }. Let xi =degG (ui ), i = 1, . . . , n, and yi =degG (vi ), i = 1, . . . , n. A subset, {ui1 , . . . , uik }, 2 ≤ k ≤ n, of V1 (G) (or {vi1 , . . . , vik } of V2 (G)) is said to be intersecting if N (ui1 )∩. . .∩N (uik ) 6= ∅ (or N (vi1 ) ∩ . . . ∩ N (vik ) 6= ∅). Let a projective plane πq of order q exist and nq = q 2 + q + 1. Let P = {p1 , . . . , pn } and L = {l1 , . . . , ln } be the point set and the line set of πq , respectively. A bipartite graph Gq with partition (P, L) is said to be the incidence point-line graph of the projective plane πq if for all i, j ∈ {1, . . . , n}, {pi , lj } is an edge of G if and only if pi ∈ lj . Let c8 (G) denote the number of 8–cycles in G. The main goal of this paper is to find a nontrivial upper bound for c8 (G), where G ∈ G6 (n, n). The results are summarized below. 2

Theorem 1. Let G ∈ G6 (n, n) be a 4–cycle free bipartite graph on 2n vertices with partition classes of size n. Then (i) c8 (G) ≤ 3 1 2

+

 n


√ 4n−3 , 2

4

−

n f (rn 4!

 − 2) , where f (t) = t(t − 1)(t − 2)(4n − 3 − 3t), rn =

and n ≥ 4.

(ii) If G has e edges and is extremal with respect to the number of 8–cycles, then rn − 2