Hamiltonicity, independence number, and pancyclicity

TL;DR

This paper improves the conditions under which a Hamiltonian graph with a given independence number is guaranteed to be pancyclic, showing that fewer vertices than previously thought suffice for this property.

Contribution

The authors establish a new bound n > c k^{7/3} for pancyclicity in Hamiltonian graphs, refining earlier results by Erdos and others.

Findings

Proved that n > c k^{7/3} ensures pancyclicity in Hamiltonian graphs.

Improved previous bounds from quadratic to sub-quadratic in k.

Demonstrated that fewer vertices are needed for pancyclicity than previously established.

Abstract

A graph on n vertices is called pancyclic if it contains a cycle of length l for all 3 \le l \le n. In 1972, Erdos proved that if G is a Hamiltonian graph on n > 4k^4 vertices with independence number k, then G is pancyclic. He then suggested that n = \Omega(k^2) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n > ck^{7/3} suffices.