Chv\'atal-Erd\H{o}s condition for pancyclicity

TL;DR

This paper extends the Chvátal-Erdős condition, showing that graphs with high connectivity relative to independence number are pancyclic, thus confirming a long-standing conjecture asymptotically.

Contribution

It generalizes the Chvátal-Erdős Hamiltonicity condition to pancyclicity, proving a conjecture by Jackson and Ordaz asymptotically.

Findings

Graphs with ppa(G) > (1+o(1)) lpha(G) are pancyclic.

Extends the Chvátal-Erd
51s condition for Hamiltonicity.

Proves a 30-year-old conjecture asymptotically.

Abstract

An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. A celebrated meta-conjecture of Bondy states that every non-trivial condition implying Hamiltonicity also implies pancyclicity (up to possibly a few exceptional graphs). We show that every graph $G$ with $\kappa(G) > (1+o(1)) \alpha(G)$ is pancyclic. This extends the famous Chv\'atal-Erd\H{o}s condition for Hamiltonicity and proves asymptotically a $30$-year old conjecture of Jackson and Ordaz.