Hamilton Paths and Cycles in Vertex-Transitive Graphs of Order $6p$

TL;DR

This paper proves that all connected vertex-transitive graphs of order 6p have Hamilton paths, and most also have Hamilton cycles, with specific exceptions related to the Petersen graph.

Contribution

It establishes the existence of Hamilton paths and cycles in vertex-transitive graphs of order 6p, extending previous results and identifying key exceptions.

Findings

All such graphs contain a Hamilton path.

Most contain a Hamilton cycle, except for a specific Petersen graph truncation.

Identifies conditions under which Hamilton cycles exist.

Abstract

It is shown that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order $6p$ which is not genuinely imprimitive contains a Hamilton cycle.