The distinguishing chromatic number of Cartesian products of two complete graphs

A labeling of a graph GG is distinguishing if it is only preserved by the trivial automorphism of GG. The distinguishing chromatic number of GG is the smallest integer kk such that GG has a distinguishing labeling that is at the same time a proper vertex coloring. The distinguishing chromatic number of the Cartesian product Kk□Kn is determined for all kk and nn. In most of the cases it is equal to the chromatic number, thus answering a question of Choi, Hartke and Kaul whether there are some other graphs for which this equality holds.