Pancyclicity of Hamiltonian and highly connected graphs

TL;DR

This paper proves that highly connected or minimum degree conditions relative to the independence number ensure pancyclicity in Hamiltonian graphs, confirming a conjecture up to a constant factor and extending classical results.

Contribution

It establishes new sufficient conditions involving connectivity and minimum degree that guarantee pancyclicity in Hamiltonian graphs, confirming a conjecture of Jackson and Ordaz up to a constant factor.

Findings

Graphs with connectivity at least 600 times their independence number are pancyclic.

Hamiltonian graphs with minimum degree at least 600 times their independence number are pancyclic.

Graphs with n at least 150 times the cube of the independence number are pancyclic.

Abstract

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest subset of the vertex set that does not contain an edge, and $\kappa(G)$ for the (vertex) connectivity, i.e. the size of the smallest subset of the vertex set that can be deleted to obtain a disconnected graph. A celebrated theorem of Chv\'atal and Erd\H{o}s says that $G$ is Hamiltonian if $\kappa(G) \ge \alpha(G)$. Moreover, Bondy suggested that almost any non-trivial conditions for Hamiltonicity of a graph should also imply pancyclicity. Motivated by this, we prove that if $\kappa(G) \ge 600\alpha(G)$ then G is pancyclic. This establishes a conjecture of Jackson and Ordaz up to a constant factor. Moreover, we obtain the more general result that if G is Hamiltonian with minimum degree $\delta(G) \ge 600\alpha(G)$ then G is pancyclic. Improving an old result of Erd\H{o}s, we also show that G is pancyclic if it is Hamiltonian and $n \ge 150\alpha(G)^3$. Our arguments use the following theorem of independent interest on cycle lengths in graphs: if $\delta(G) \ge 300\alpha(G)$ then G contains a cycle of length $\ell$ for all $3 \le \ell \le \delta(G)/81$.