## 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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### References:

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