Trivial coloring of Cartesian product of graphs

Abstract

A coloring of a direct product of graphs is said to be {\em trivial} iff it is induced by some coloring of a factor of the product. A graph $G$ is trivially power colorable iff every coloring of a finite power of $G$ with $\chi (G)$-many colors is trivial. Greenwell and Lovász proved that the finite complete graphs $K_n$ for $n\ge 3$ are trivially power colorable. Generalizing their result we define a much wider class of trivially power-colorable graphs: if $G$ is a finite, connected graph with $\chi (G)\ge 3$ and every vertex of $G$ is in a clique of size $\chi (G)$, then $G$ is trivially power-colorable. As an application of this result, we give a complete characterization of trivially power-colorable cographs. Finally, we give a structural description of the colorings of infinite powers of trivially power-colorable finite graphs.