Pancyclicity of highly connected graphs

TL;DR

This paper proves that large graphs with vertex connectivity greater than independence number are pancyclic, confirming a conjecture for large graphs and extending classical Hamiltonicity results to cycle richness.

Contribution

It establishes that graphs with high connectivity relative to independence number are pancyclic, confirming a longstanding conjecture for large graphs and improving recent results.

Findings

Graphs with $ abla(G) > eta(G)$ are pancyclic for large $|G|$

Confirms Jackson and Ordaz's conjecture for large graphs

Improves upon recent results by Draganić, Munhá-Correia, and Sudakov

Abstract

A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a graph $G$ satisfies $\kappa(G) \ge \alpha(G)$, where $\kappa(G)$ is the vertex-connectivity of $G$, then $G$ has a Hamilton cycle. We prove a similar result implying that a graph $G$ is pancyclic, namely it contains cycles of all lengths between $3$ and $|G|$: if $|G|$ is large and $\kappa(G) > \alpha(G)$, then $G$ is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990) for large graphs, and improves upon a very recent result of Dragani\'c, Munh\'a-Correia, and Sudakov.