An Ordered Turán Problem for Bipartite Graphs

Let F be a graph. A graph G is F -free if it does not contain F as a subgraph. The Turán number of F , written ex(n, F ), is the maximum number of edges in an F -free graph with n vertices. The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem. In this paper we introduce an ordered version of the Turán problem for bipartite graphs. Let G be a graph with V (G) = {1, 2, . . . , n} and view the vertices of G as being ordered in the natural way. A zig-zag Ks,t, denoted Zs,t, is a complete bipartite graph Ks,t whose parts A = {n1 < n2 < · · · < ns} and B = {m1 < m2 < · · · < mt} satisfy the condition ns < m1. A zig-zag C2k is an even cycle C2k whose vertices in one part precede all of those in the other part. Write Z2k for the family of zig-zag 2k-cycles. We investigate the Turán numbers ex(n,Zs,t) and ex(n,Z2k). In particular we show ex(n,Z2,2) 6 2 3n 3/2 + O(n5/4). For infinitely many n we construct a Z2,2-free nvertex graph with more than (n− √ n− 1) + ex(n,K2,2) edges.