Hamilton cycles in vertex-transitive graphs of order $6p$

TL;DR

This paper proves that all connected vertex-transitive graphs of order 6p, with p prime, contain Hamilton cycles, except for one specific graph related to the Petersen graph, extending previous results on Hamilton paths.

Contribution

It establishes the existence of Hamilton cycles in vertex-transitive graphs of order 6p, identifying a unique exception, thus advancing understanding of Hamiltonicity in symmetric graphs.

Findings

All such graphs contain Hamilton cycles

The Petersen graph exception is unique

Extends previous Hamilton path results

Abstract

It was shown by Kutnar and \v Sparl in 2009 that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. In this paper, it will be shown that every such graph contains a Hamilton cycle, except for the triangle-replaced graph of the Petersen graph.