The distinguishing number of Cartesian products of complete graphs

Abstract: The distinguishing number of a graph is the least integer such that has a labeling with labels that is preserved only by a trivial automorphism. We prove that Cartesian products of relatively prime graphs whose sizes do not differ too much can be distinguished with a small number of colors. We determine the distinguishing number of the Cartesian product for all and , either explicitly or by a short recursion. We also introduce column-invariant sets of vectors and prove a switching lemma that plays a key role in the proofs. [Copyright &y& Elsevier]