The Lov\'asz conjecture holds for moderately dense Cayley graphs

TL;DR

This paper proves that large, moderately dense Cayley graphs with degree at least a polynomial function of their size contain Hamilton cycles, advancing the Lovász conjecture without using Szemerédi's regularity lemma.

Contribution

It establishes a new degree condition for Hamiltonicity in Cayley graphs, improving previous results and employing an arithmetic regularity lemma tailored to Cayley graphs.

Findings

Large Cayley graphs with degree d ≥ n^{1-c} have Hamilton cycles.

Progress towards the Lovász conjecture for dense Cayley graphs.

Avoids Szemerédi's regularity lemma by using an arithmetic regularity lemma.

Abstract

We show that there is an absolute constant $c>0$ such that every large connected $n$-vertex Cayley graph with degree $d\geq n^{1-c}$ has a Hamilton cycle. This makes progress towards the Lov\'asz conjecture and improves upon the previous best result of this form due to Christofides, Hladk\'y, and M\'ath\'e from 2014 concerning graphs with $d\geq \varepsilon n$. Our proof avoids the use of Szemer\'edi's regularity lemma and relies instead on an efficient arithmetic regularity lemma specialised to Cayley graphs.