Star chromatic number of some graph products

Abstract

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.