On products of graphs and regular groups

A graph X is called a graphical regular representation (GRR) of a group G if the automorphism group of X is regular and isomorphic to G. Watkins and Nowitz have shown that the direct product G×H of two finite groups G and H has a GRR if both factors have a GRR and if at least one factor is different from the cyclic group of order two. We give a new proof of this result, thereby removing the restriction to finite groups. We further show that the complement X′ of a finite or infinite graph X is prime with respect to cartesian multiplication if X is composite and not one of six exceptional graphs.