Cartesian powers of graphs can be distinguished by two labels

The distinguishing number D ( G ) of a graph G is the least integer d such that there is a d -labeling of the vertices of G which is not preserved by any nontrivial automorphism. For a graph G let G r be the r th power of G with respect to the Cartesian product. It is proved that D ( G r ) = 2 for any connected graph G with at least 3 vertices and for any r ≥ 3 . This confirms and strengthens a conjecture of Albertson. Other graph products are also considered and a refinement of the Russell and Sundaram motion lemma is proved.