Star coloring of sparse graphs

Abstract

A proper coloring of the vertices of a graph is called a \emph{star coloring} if the union of every two color classes induces a star forest. The star chromatic number ${\chi }_s(G)$ 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 ${\chi }_s(G)$ and the maximum average degree $\mbox{Mad}(G)$ of a graph $G$. We prove that: (1) If $G$ is a graph with $\mbox{Mad}(G) < \frac{26}{11}$, then ${\chi }_s(G)\le 4$. (2) If $G$ is a graph with $\mbox{Mad}(G) < \frac{18}{7}$ and girth at least 6, then ${\chi }_s(G)\le 5$. (3) If $G$ is a graph with $\mbox{Mad}(G) < \frac{8}{3}$ and girth at least 6, then ${\chi }_s(G)\le 6$. These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets.

20 pages, 5 figures