Star Coloring of Certain Graph Classes

Abstract

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, denoted by $$\chi _s(G)$$ , is the minimum number of colors needed to star color G. Similar to the notion of $$\chi $$ -boundedness in graphs, we say that a hereditary class of graphs $$\mathcal {G}$$ is $$\chi _s$$ -bounded if, for some integer valued function f, $$\chi _s(G) \le f(\omega (G))$$ for every $$G \in \mathcal {G}$$ , where $$\omega (G)$$ is the maximum number of vertices that are pairwise adjacent in G. We show that some classes of perfect graphs, some classes of ( $$P_5, C_4$$ )-free graphs, and some classes of $$K_{1, 3}$$ -free graphs are $$\chi _s$$ -bounded. We also give examples in most of the cases to show that the bounds are tight.