Two counterexamples to a conjecture about even cycles

TL;DR

This paper presents two counterexamples to Verstra"ete's conjecture on the existence of large $C_{2\, ext{ell}}$-free subgraphs in $C_{2k}$-free graphs, disproving it for specific parameters.

Contribution

The paper provides the first known counterexamples to the conjecture for the case $\, ext{ell}=4$ and $k=5$, using constructions from hypercube graphs and Wenger's extremal graphs.

Findings

Counterexamples for $\, ext{ell}=4$, $k=5$ disproving the conjecture.

Uses dense $C_{10}$-free subgraph of the hypercube.

Employs Wenger's extremal $C_{10}$-free graph construction.

Abstract

A conjecture of Verstra\"ete 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 K\"uhn 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.