Weakly pancyclic vertices in dense nonbipartite graphs

TL;DR

This paper proves that dense nonbipartite graphs with certain size conditions contain at least three vertices that lie on cycles of all lengths between the girth and circumference, strengthening previous results.

Contribution

It establishes the existence of three weakly pancyclic vertices in dense nonbipartite graphs, improving upon earlier work by Brandt from 1997.

Findings

At least three weakly pancyclic vertices in dense nonbipartite graphs.

A specific size threshold guarantees the existence of these vertices.

An exception case is identified where the result does not hold.

Abstract

Let $G$ be a graph of girth $g$ and circumference $c.$ A vertex $v$ of $G$ is called weakly pancyclic if $v$ lies on an $\ell$-cycle for every integer $\ell$ with $g\le \ell\le c.$ We prove that if $G$ is a nonbipartite graph of order $n\ge 5$ and size at least $\left\lfloor(n-1)^2/4\right\rfloor+2,$ then $G$ contains three weakly pancyclic vertices, with one exception. This strengthens a result of Brandt from 1997. We also pose a related problem.