Star chromatic bounds

A star coloring of an undirected graph G is a coloring of the vertices of G such that (i) no two adjacent vertices receive the same color, and (ii) no path on four vertices (not necessarily induced) is bi-colored. The star chromatic number of G is the minimum number of colors needed to star color G. In this note, we deduce upper bounds for the star chromatic number in terms of the clique number for some special classes of graphs which are defined by forbidden induced subgraphs.