Star coloring of sparse graphs

A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induce a star forest. The star chromatic number χs(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chromatic number χs(G) and the maximum average degree Mad(G) of a graph G. We prove that: 1. If G is a graph with Mad(G) < 26 11 , then χs(G) ≤ 4. 2. If G is a graph with Mad(G) < 18 7 and girth at least 6, then χs(G) ≤ 5. 3. If G is a graph with Mad(G) < 8 3 and girth at least 6, then χs(G) ≤ 6. These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets.