Pancyclicity of Hamiltonian graphs

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 $n$. In 1972, Erdős conjectured that every Hamiltonian graph with independence number at most $k$ and at least $n=\Omega (k^2)$ vertices is pancyclic. We prove this old conjecture in a strong form by showing that if such a graph has $n=(2+o(1))k^2$ vertices, it is already pancyclic, and this bound is asymptotically best possible.