Hamilton Cycles In Vertex-Transitive Graphs of Order 10p

TL;DR

This paper proves that all connected vertex-transitive graphs of order 10p, with p prime, contain a Hamilton cycle except for the truncated Petersen graph, extending previous results on Hamiltonicity in such graphs.

Contribution

It establishes the Hamilton cycle existence in connected vertex-transitive graphs of order 10p for all primes p, except for a specific known exception.

Findings

All such graphs contain a Hamilton cycle.

The only exception is the truncation of the Petersen graph.

Extends previous Hamiltonicity results for graphs of order 10p.

Abstract

After long-term efforts, the Hamilton path (cycle) problem for connected vertex-transitive graphs of order $pq$ (where $p$ and $q$ are primes) was finally resolved in 2021, see [10]. Fifteen years ago, mathematicians began addressing this problem for graphs of order $2pq$. Among these studies, it was proved in 2012 (see [21]) that every connected vertex-transitive graph of order $10p$ (where $p \neq 7$ is a prime) contains a Hamilton path, with the exception of a family of graphs that was recently confirmed in [11]. In this paper, we achieve a further result: every connected vertex-transitive graph of order $10p$ (where $p$ is a prime) contains a Hamilton cycle, except for the truncation of the Petersen graph.