Distinguishing Cartesian powers of graphs

The distinguishing number D(G) of a graph is the least integer d such that there is a d-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G 6= K2,K3 with respect to the Cartesian product is 2. This result strengthens results of Albertson [1] on powers of prime graphs, and results of Klavžar and Zhu [14]. More generally, we also prove that d(G¤H) = 2 if G and H are relatively prime and |H| ≤ |G| < 2|H| − |H|. Under additional conditions similar results hold for strong and direct products.