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

Abstract

A retract of a graph $\Gamma$ is an induced subgraph $\Psi$ of $\Gamma$ such that there exists a homomorphism from $\Gamma$ to $\Psi$ whose restriction to $\Psi$ 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 $\Gamma$ is $G$-symmetric if $G$ is a subgroup of the automorphism group of $\Gamma$ 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 $\Gamma$ admits a nontrivial partition that is preserved by $G$, then $\Gamma$ is an imprimitive $G$-symmetric graph. In this paper cores of imprimitive symmetric graphs $\Gamma$ of order a product of two distinct primes are studied. In many cases the core of $\Gamma$ is determined completely. In other cases it is proved that either $\Gamma$ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.