The dominated coloring (dom-coloring) of a graph G is a proper coloring such that each color class is dominated by at least one vertex. The dominated chromatic number (dom-chromatic number) of G is the minimum number of color classes among all dominated colorings of G, denoted by \(\chi _{\text {dom}}(G)\). In this paper, we study the dominated coloring of Cartesian product, direct product, lexicographic product and strong product of some graphs.

中文翻译：

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产品图表中的主导颜色

图G的主导着色（dom-coloring）是一种适当的着色，使得每个颜色类由至少一个顶点主导。G的主色数（dom-色数）是G的所有主着色中颜色类的最小数量，用\(\chi _{\text {dom}}(G)\)表示。本文研究了一些图的笛卡尔积、直积、词典积和强积的主导着色。