Distinguishing Chromatic Number of Cartesian Products of Graphs

The distinguishing chromatic number χD (G) of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number χ(·). Our main result is that for every graph G, there exists a constant dG such that for all d ≥ dG the distinguishing chromatic number of Gd is at most χ(G)+1, where Gd is the Cartesian product of d copies of G. We also prove that for d ≥ 5, the Cartesian product of d complete graphs has distinguishing chromatic number at most one more than the corresponding chromatic number, and we determine the distinguishing chromatic number of hypercubes exactly.