Two counterexamples to a conjecture about even cycles

Abstract

A conjecture of Verstraëte states that for any fixed $\ell <k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell }$-free subgraph with at least $c|E(G)|$ edges. For $\ell =2$, this conjecture was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for $\ell =4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger's construction for extremal $C_{10}$-free graphs.

7 pages; this paper replaces arXiv:2501.13036 (which will not be published)