Geodesic
2-Tone Colorings in Graph Products
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 55-72

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A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set 1, . . ., k, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a t- tone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in three different graph products. In particular, given graphs G and H, we bound the 2-tone chromatic number for the direct product G×H, the Cartesian product G□H, and the strong product G⊠H.
Keywords: t-tone coloring, Cartesian product, direct product, strong product 
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Loe, Jennifer; Middelbrooks, Danielle; Morris, Ashley; Wash, Kirsti. 2-Tone Colorings in Graph Products. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 55-72. http://geodesic.mathdoc.fr/item/DMGT_2015_35_1_a4/