Pancyclicity of highly connected graphs

Abstract

A well-known result due to Chvátal and Erdős (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, that is, 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 Draganić, Munhá Correia, and Sudakov.