2016•Discussiones Mathematicae Graph TheoryOpen access

Edge-transitive lexicographic and Cartesian products

Wilfried Imrich, Ali Iranmanesh, Sandi Klavžar, Abolghasem Soltani

Abstract

In this note connected, edge-transitive lexicographic and Cartesian products are characterized. For the lexicographic product G H of a connected graph G that is not complete by a graph H, we show that it is edge-transitive if and only if G is edge-transitive and H is edgeless. If the first factor of G H is non-trivial and complete, then G H is edge-transitive if and only if H is the lexicographic product of a complete graph by an edgeless graph. This fixes an error of Li, Wang, Xu, and Zhao [11]. For the Cartesian product it is shown that every connected Cartesian product of at least two non-trivial factors is edge-transitive if and only if it is the Cartesian power of a connected, edge-and vertex-transitive graph.

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Key concepts: Cartesian product, Lexicographical order, Transitive relation, Mathematics, Transitive reduction, Combinatorics, Transitive closure, Vertex (graph theory)