On Vertex-Transitive Graphs of Order $qp$

Abstract

The structure and the hamiltonicity of vertex-transitive graphs of order $qp$, where $q$ and $p$ are distinct primes, are studied. It is proved that if $q<p$ and $\text{p}\not\equiv 1(\mathrm{mod}\text{q})$ and $G$ is a vertex-transitive graph of order $qp$ such that $AutG$ contains an imprimitive subgroup, then either $G$ is a circulant or $V(G)$ partitions into $p$ subsets of cardinality $q$ such that there exists a perfect matching between any two of them. Partial results are obtained for $\text{p}\equiv 1(\mathrm{mod}\text{q})$. Moreover, it is proved that every connected vertex-transitive graph of order $3p$ is hamiltonian.