On two cycles of consecutive even lengths

TL;DR

This paper proves that graphs with a certain average degree contain two cycles of consecutive even lengths, extending previous results and solving a special case of a conjecture, with tight bounds and structural analysis.

Contribution

It establishes a new average degree condition ensuring the existence of two consecutive even cycles, solving a special case of Verstra"ete's conjecture with tight bounds.

Findings

Graphs with at least 2.5(n-1) edges contain two consecutive even cycles.

The result is tight and provides the extremal number for cycles of length two mod four.

The proof involves structural analysis and applies to 3-connected graphs.

Abstract

Bondy and Vince showed that every graph with minimum degree at least three contains two cycles of lengths differing by one or two.We prove the following average degree counterpart that every $n$-vertex graph $G$ with at least $\frac52(n-1)$ edges, unless $4|(n-1)$ and every block of $G$ is a clique $K_5$, contains two cycles of consecutive even lengths. Our proof is mainly based on structural analysis, and a crucial step which may be of independent interest shows that the same conclusion holds for every 3-connected graph with at least 6 vertices. This solves a special case of a conjecture of Verstra\"ete. The quantitative bound is tight and also provides the optimal extremal number for cycles of length two modulo four.