Weakly pancyclic vertices in dense nonbipartite graphs

TL;DR

This paper strengthens Brandt’s result by proving that every nonbipartite graph of order $n\ge5$ with size at least $\left\lfloor(n-1)^2/4\right\rfloor+2$, except for the special graph $BT(n)$, contains three weakly pancyclic vertices. The authors prove this via an inductive framework that treats both hamiltonian and nonhamiltonian cases, employing structural lemmas about small/big vertices and dense balanced bipartite subgraphs to construct cycles of all lengths between the girth and circumference. A sharpness result is provided by constructing graphs with exactly three weakly pancyclic vertices, and the paper closes with related open problems, including a conjecture on pancyclic edges and a function $f(n)$ for minimal edge-boundaries guaranteeing a weakly pancyclic vertex. The results deepen understanding of cycle-rich structure in dense nonbipartite graphs and delineate the exceptional role of $BT(n)$ in such phenomena.

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.

Figures (3)

• Figure 1: Weakly pancyclic graphs with no weakly pancyclic vertex
• Figure 2: The graphs $BT(8)$ and $BT(9)$
• Figure 3: The graphs $G_9$ and $G_{10}$