Size conditions for admissible or consecutive even cycles in graphs

TL;DR

This paper extends size conditions for the existence of certain cycles in graphs, characterizing extremal graphs and proving a conjecture related to graphs with no consecutive even cycles.

Contribution

It provides a sharp size analogue of a previous cycle existence result and proves Verstra"ete's conjecture for a specific range of graph sizes.

Findings

Characterized extremal graphs attaining the lower bound for admissible cycles.

Proved Verstra"ete's conjecture for 2k+2 ≤ n ≤ 4k+1.

Obtained a stronger result in the specified size range.

Abstract

In 2022, Gao, Huo, Liu, and Ma proved that every graph with minimum degree at least $k+1$ contains $k$ admissible cycles, where a set of $k$ cycles is said to be admissible if their lengths form an arithmetic progression with common difference one or two. In this paper, we provide a sharp size analogue of their result and characterize the extremal graphs attaining the lower bound. In 2016, Verstra\"ete conjectured that every $n$-vertex graph $G$ containing no $k$ cycles of consecutive even lengths has at most $(2k+1)(n-1)/2$ edges, with equality only if every block of $G$ is a clique of order $2k+1$. We prove this conjecture for $2k+2\leq n\leq 4k+1$, and in fact obtain a stronger result in this range.