On Vertex-Imprimitive Graphs of Order a Product of Three Distinct Odd Primes

Abstract

This paper contributes to the determination of all integers of the form $pqr$, where $p$, $q$, and $r$ are distinct odd primes, for which there exists a vertex-transitive graph on $pqr$ vertices that is not a Cayley graph. The paper addresses the situation where there exists a vertex-transitive subgroup $G$ of automorphisms of such a graph
which has a chain $1<N<K<G$ of normal subgroups, such that both $N$ and $K$ are intransitive on vertices and the $N$-orbits are proper subsets of the $K$-orbits.