Cartesian powers of graphs can be distinguished by two labels

Abstract

AbstractThe 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 Gr be the rth power of G with respect to the Cartesian product. It is proved that D(Gr)=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