On the maximum number of C ′ 6 s in a quadrilateral - free bipartite graph GENE

Let G = G(n, n) be a 4–cycle free, bipartite graph on 2n vertices with partitions of equal cardinality n. Let c6(G) denote the number of cycles of length 6 in G. We prove that for n ≥ 4, c6(G) ≤ 13 ( n 2 ) (n − rn), where rn = 12 + √ 4n−3 2 , with equality if and only if G is the incidence point-line graph of a projective plane. 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 Turán Theorem. For the same H, if G is the family of all (m,n)– bipartite graphs, 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]. We start with the following definitions and notation. All missing ones can be found in [2]. LetG = G(m,n) be a bipartite graph onm+n vertices with partition (V1(G), V2(G)) such that V1(G) = {u1, . . . , um}, V2(G) = {v1, . . . , vn}. Denote the number of edges of a graph G by e = e(G), the neighborhood of a vertex v ∈ V (G) by N(v) (v 6∈ N(v)), and the degree of vertex v in G by degG(v). Let xi =degG(ui), i = 1, . . . ,m, 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 ifN(ui1)∩. . .∩N(uik) 6= ∅ (or N(vi1) ∩ . . . ∩N(vik) 6= ∅). For a graph G containing a cycle, the girth of G is the length of a shortest cycle in G. Let n = q+q+1 and let π be a finite projective plane of order q with point set P = {p1, . . . , pn} and line set L = {l1, . . . , ln}. A bipartite graph G with partitions (P,L) is said to be the incidence point-line graph of the projective plane π if for all i, j ∈ {1, . . . , n}, {pi, lj} is an edge of G if and only if pi ∈ lj . Let c6(G) denote the number of 6–cycles in G. The main goal of this paper is to find a nontrivial upper bound for c6(G), where G = G(m,n) is a bipartite 4–cycle free graph. The results are summarized below. 2 Theorem 1. Let G be a 4–cycle free bipartite graph on m+n vertices with partition classes of size m and n. Then for m ≥ n ≥ 4, c6(G) ≤ m3 ( rm,n 2 ) (n − rm,n), where rm,n = 1 2m [m+ (m 2 + 4mn(n− 1)) 2 ]. In the case when m = n we can actually say much more. Theorem 2. Let G be a 4–cycle free bipartite graph on 2n vertices with partitions of size n. Then for n ≥ 4, c6(G) ≤ 13 ( n 2 ) (n− rn), where rn = 12 + 1 2 √ 4n− 3, with equality if and only if G is the incidence point-line graph of a projective plane. Using the terminology of finite geometries (see [1]), Theorems 1 and 2 provide an upper bound for the number of triangles in near–linear spaces with m points and n lines. The paper is organized in the following way. In Section 2 we prove Theorem 1. In Section 3 we present the original proof of Theorem 2 as a corollary of Theorem 1, and sketch another proof, independent of Theorem 1, which is based on a recent result in [7]. Section 2: General case Let G = G(m,n) be a bipartite, 4–cycle free graph on m + n vertices with partition (V1(G), V2(G)) such that V1(G) = {u1, . . . , um}, V2(G) = {v1, . . . , vn}. Let xi =degG(ui) and yi =degG(vi), i = 1, . . . , n. It is clear that m ∑ i=1 ( xi 2 ) ≤ ( n 2 ) (see [2, ch. VI.2] for a more general result). For positive integers m and n, define rm,n = 1 2m [m + (m 2 + 4mn(n − 1)) 1 2 ]. Obviously, we may assume that degG(v) ≥ 1 for all v ∈ V (G). 3 To obtain a bound on the number of cycles of length six in G we introduce a real–valued function in IR which will provide an upper estimate of c6(G) and we show that this function attains its maximum on the hypersphere defined by m ∑ i=1 ( xi 2 ) = ( n 2 ) . Then we prove that this maximum occurs at the point r = (rm,n, · · · , rm,n), where rm,n = 1 2m [m+ (m 2 + 4mn(n− 1)) 2 ]. Let I = [1, n] ⊂ IR and I be the Cartesian Product of m copies of I. Let Sm,n be the region in IR defined by Sm,n = { x ∈ I ∣∣∣∣ m ∑ i=1 ( xi 2 ) ≤ ( n 2 )} . Define the function F on Sm,n by F (x) = m ∑ i=1 f(xi) , where f is defined by f(x) =