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

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.