This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.
A star coloring of a graph [Formula: see text] is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors have connected components that are star graphs. A graph [Formula: see text] is [Formula: see text]- star-colorable if there exists a star coloring of [Formula: see text] from a set of [Formula: see text] colors. The minimum positive integer [Formula: see text] for which [Formula: see text] is [Formula: see text]-star-colorable is the star chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, upper and lower bounds are presented for the star chromatic number of the rooted product, hierarchical product, and lexicographic product.
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