The distinguishing chromatic number χ D (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 d G such that for all d ≥ d G the distinguishing chromatic number of G d is at most χ(G)+ 1, where G d 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.

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A d-block is a 0, 1-matrix in which every row has sum d. Let S n be the set of pairs (k, l) such that the columns of any (k + l)-block with n rows split into a k-block and an l-block. For n ≥ 5, we prove the general necessary condition that (k, l) ∈ S n only if each element of {1, . . . , n} divides k or l. We also determine S n for n ≤ 5. Trivially,: 6 | kl and min{k, l} > 1}, and S 5 = {(k, l) : 3, 4, 5 each divide k or l, plus 11 = min{k, l} > 7}.