On Hamilton paths in vertex-transitive graphs of order $10p$

TL;DR

This paper investigates Hamilton cycles in specific vertex-transitive graphs of order 10p, extending previous work by identifying Hamilton cycles in previously excluded exceptional cases.

Contribution

It provides a proof of the existence of Hamilton cycles in the exceptional vertex-transitive graphs of order 10p that were not covered in earlier research.

Findings

Hamilton cycles are found in the previously excluded exceptional graphs.

The paper confirms the presence of Hamilton cycles in all vertex-transitive graphs of order 10p.

Extends understanding of Hamiltonicity in symmetric graphs.

Abstract

It was shown by Kutnar, Maru\v si\v c and Zhang in 2012 that every connected vertex-transitive graph of order $10p$, where $p$ is a prime and $p\ne 7$, contains a Hamilton path, except for graphs $X$ arising from the action of PSL$(2, s^m)$ on cosets of $\mathbb{Z}_s^m\rtimes \mathbb{Z}_{\frac{s^m-1}{10}}$, where $s$ is a prime. In this paper, Hamilton cycles of these exceptions $X$ will be found.