Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph i¾? is an induced subgraph Ψ of i¾? such that there exists a homomorphism from i¾? 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 i¾? is G-symmetric if G is a subgroup of the automorphism group of i¾? 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 i¾? admits a nontrivial partition that is preserved by G, then i¾? is an imprimitive G-symmetric graph. In this paper cores of imprimitive symmetric graphs i¾? of order a product of two distinct primes are studied. In many cases the core of i¾? is determined completely. In other cases it is proved that either i¾? is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.

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