Cycles of many lengths in Hamiltonian graphs

TL;DR

This paper proves that Hamiltonian graphs with minimum degree at least 3 contain nearly linear many distinct cycle lengths, resolving longstanding conjectures in graph theory.

Contribution

It confirms that such graphs have at least n^{1-o(1)} different cycle lengths, improving previous bounds from square root to nearly linear.

Findings

Number of cycle lengths in these graphs is at least n^{1-o(1)}

Resolved conjectures from Jacobson-Lehel and Verstraëte asymptotically

Improved lower bounds from √n to nearly linear in n

Abstract

In 1999, Jacobson and Lehel conjectured that for $k \geq 3$, every $k$-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstra\"{e}te, who asked whether the regularity can be replaced with the weaker condition that the minimum degree is at least $3$. Despite attention from various researchers, until now, the best partial result towards both of these conjectures was a $\sqrt{n}$ lower bound on the number of cycle lengths. We resolve these conjectures asymptotically, by showing that the number of cycle lengths is at least $n^{1-o(1)}$.