The core of a vertex transitive complementary prism of a lexicographic product

Abstract

The complementary prism of a graph Γ is the graph ΓΓ̄, which is formed from the union of Γ and its complement Γ̄ by adding an edge between each pair of identical vertices in Γ and Γ̄. Vertex-transitive self-complementary graphs provide vertex-transitive complementary prisms. It was recently proved by the author that ΓΓ̄ is a core, i.e. all its endomorphisms are automorphisms, whenever Γ is vertex-transitive, self-complementary, and either Γ is a core or its core is a complete graph. In this paper the same conclusion is obtained for some other classes of vertex-transitive self-complementary graphs that can be decomposed as a lexicographic product Γ = Γ₁[Γ₂]. In the process some new results about the homomorphisms of a lexicographic product are obtained.