Star coloring of Tensor product of two graphs

A star coloring of a graph [Formula: see text] is a proper vertex coloring such that no path on four vertices is bicolored. The smallest integer [Formula: see text] for which [Formula: see text] admits a star coloring with [Formula: see text] colors is called the star chromatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we study the star coloring of the tensor product of two graphs and obtain the following results. (1) We give an upper bound on the star chromatic number of the tensor product of two arbitrary graphs. (2) We determine the exact value of the star chromatic number of the tensor product of two paths. (3) We show that the star chromatic number of the tensor product of two cycles is five, except for [Formula: see text] and [Formula: see text]. (4) We give tight bounds for the star chromatic number of the tensor product of a cycle and a path.