Cartesian powers of graphs can be distinguished by two labels

The distinguishing number D(G)D(G) of a graph GG is the least integer dd such that there is a dd-labeling of the vertices of GG which is not preserved by any nontrivial automorphism. For a graph GG let GrGr be the rrth power of GG with respect to the Cartesian product. It is proved that D(Gr)=2D(Gr)=2 for any connected graph GG with at least 3 vertices and for any r≥3r≥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.