Hamilton Cycles In Primitive Graphs of Order $2rs$

TL;DR

This paper proves that, under primitive automorphism group action, all connected vertex-transitive graphs of order 2rs (except Coxeter graph) contain Hamilton cycles, extending known results for other orders.

Contribution

It establishes the existence of Hamilton cycles in primitive, vertex-transitive graphs of order 2rs, filling a gap in the Hamiltonian problem for such graph orders.

Findings

All such graphs (except Coxeter) have Hamilton cycles.

The automorphism group acting primitively is key to the result.

Extends previous results from order rs to 2rs.

Abstract

After long term efforts, it was recently proved in \cite{DKM2} that except for the Peterson graph, every connected vertex-transitive graph of order $rs$ has a Hamilton cycle, where $r$ and $s$ are primes. A natural topic is to solve the hamiltonian problem for connected vertex-transitive graphs of $2rs$. This topic is quite trivial, as the problem is still unsolved even for that of $r=3$. In this paper, it is shown that except for the Coxeter graph, every connected vertex-transitive graph of order $2rs$ contains a Hamilton cycle, provided the automorphism group acts primitively on vertices.