Pancyclicity of Hamiltonian graphs

TL;DR

This paper proves a longstanding conjecture that large enough Hamiltonian graphs with bounded independence number are pancyclic, establishing the precise asymptotic bound for the number of vertices needed.

Contribution

It confirms Erd ext{"o}s's conjecture by showing that Hamiltonian graphs with n = (2+o(1))k^2 vertices and independence number at most k are pancyclic, with the bound being asymptotically optimal.

Findings

Proves Erd ext{"o}s's conjecture in a strong form.

Establishes the asymptotically best possible bound for pancyclicity.

Shows that graphs with n = (2+o(1))k^2 vertices are pancyclic under the given conditions.

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, Erd\H{o}s conjectured that every Hamiltonian graph with independence number at most $k$ and at least $n = \Omega(k^2)$ vertices is pancyclic. In this paper 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.