Even cycles in graphs avoiding longer even cycles

TL;DR

The paper investigates a conjecture about the existence of dense subgraphs avoiding certain even cycles in graphs, confirming it for some cases and providing counterexamples for others.

Contribution

It proves the conjecture holds for specific cycle lengths and constructs counterexamples for others, advancing understanding of cycle-avoidance in graphs.

Findings

Confirmed the conjecture for C6 and all C_{2k} with odd k.

Counterexamples found for C8 and C10 using hypercube constructions.

Identified limitations of the conjecture in certain cycle-avoidance scenarios.

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. 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 $C_{10}$.