Pancyclicity of Hamiltonian and highly connected graphs

Peter Keevash, Benny Sudakov

Published 2009-03-26Version 1

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 $3 \le \ell \le \delta(G)/81$.