2-Tone Colorings in Graph Products

• Main Authors: Loe Jennifer, Middelbrooks Danielle, Morris Ashley, Wash Kirsti
• Format: Article
• Language: English
• Published: Sciendo 2015-02-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: t-tone coloring; cartesian product; direct product; strong product
• Online Access: https://doi.org/10.7151/dmgt.1773

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.