Cores of imprimitive symmetric graphs of order a product of two distinct primes

Abstract

A retract of a graph $Γ$ is an induced subgraph $Ψ$ of $Γ$ such that there exists a homomorphism from $Γ$ to $Ψ$ whose restriction to $Ψ$ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph $Γ$ is $G$-symmetric if $G$ is a subgroup of the automorphism group of $Γ$ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of $Γ$ admits a nontrivial partition that is preserved by $G$, then $Γ$ is an imprimitive $G$-symmetric graph. In this paper cores of imprimitive symmetric graphs $Γ$ of order a product of two distinct primes are studied. In many cases the core of $Γ$ is determined completely. In other cases it is proved that either $Γ$ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.