Star coloring of subcubic graphs

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 4 vertices is bi-colored. The star chromatic number of G, Xs(G), is the minimum number of colors needed to star color G. In this paper, we show that if a graph G is either non-regular subcubic or cubic with girth at least 6, then Xs(G) ≤ 6, and the bound can be realized in linear time.