On subgraphs of C_2k-free graphs and a problem of Kühn and Osthus

Abstract

Let c denote the largest constant such that every C 6 -free graph G contains a bipartite and C 4 -free subgraph having c fraction of edges of G. Győri et al. showed that 3 8 ≤ c ≤ 2 5 . We prove that c = 3 8 . More generally, we show that for any ε > 0, and any integer k ≥ 2, there is a C 2k -free graph G 1 which does not contain a bipartite subgraph of girth greater than 2k with more than 1 − 1 2 2k−2 2 2k−1 (1 + ε) fraction of the edges of G 1 . There also exists a C 2k -free graph G 2 which does not contain a bipartite and C 4 -free subgraph with more than 1 − 1 2 k−1

On the Turán number of some ordered even cycles

“…Our proof of Theorem 4 is inspired by the proof of Grósz, Methuku and Tompkins in [11] on deleting 4-cycles, which is a simple proof of a theorem of Kühn and Osthus [13]. We make use of the following result of Gallai [10] and Roy [17].…”