Even cycles in graphs avoiding longer 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. We show that $C_6$ and $C_{2k}$ satisfy the conjecture for all odd $k$, but observe that a recent construction of a dense $C_{10}$-free subgraph of the hypercube yields a counterexample to the conjecture for $C_8$ and $C_{10}$.

Our main result follows from the work of Kühn and Osthus [J. Graph Theory 48 (2005), 147--156]. In particular, our Theorem 1.4 can be proved using their Lemma 10, with g=6