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The distinguishing chromatic number of Cartesian products of two complete graphs. (English) Zbl 1222.05225

Summary: A labeling of a graph \(G\) is distinguishing if it is only preserved by the trivial automorphism of \(G\). The distinguishing chromatic number of \(G\) is the smallest integer \(k\) such that \(G\) has a distinguishing labeling that is at the same time a proper vertex coloring. The distinguishing chromatic number of the Cartesian product \(K_k\square K_n\) is determined for all \(k\) and \(n\). 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.

MSC:

05C76 Graph operations (line graphs, products, etc.)
05C15 Coloring of graphs and hypergraphs
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