Star coloring of sparse graphs

Abstract

Abstract A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induces a star forest. The star chromatic number χ s ( 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 χ s ( G ) and the maximum average degree Mad ( G ) of a graph G . We prove that: If G is a graph with , then χ s ( G )≤4. If G is a graph with and girth at least 6, then χ s ( G )≤5. If G is a graph with and girth at least 6, then χ s ( G )≤6. These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 201–219, 2009