On star 5-colorings of sparse graphs

Abstract

A stark-coloring of a graph G is a proper (vertex) k-coloring of G such that the vertices on a path of length three receive at least three colors. Given a graph G, its star chromatic number, denoted χs(G), is the minimum integer k for which G admits a star k-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree of a graph; the maximum average degree, denoted mad(G), of a graph G is max2|E(H)||V(H)|:H⊂G. It is known that for a graph G, if mad(G)<83, then χs(G)≤6 (Kündgen and Timmons, 2010), and if mad(G)<187 and its girth is at least 6, then χs(G)≤5 (Bu et al., 2009). We improve both results by showing that for a graph G, if mad(G)≤83, then χs(G)≤5. As an immediate corollary, we obtain that a planar graph with girth at least 8 has a star 5-coloring, improving the best known girth condition for a planar graph to have a star 5-coloring (Kündgen and Timmons, 2010; Timmons, 2008).