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 $\chi_s(G)\leq 4$. (2) If $G$ is a graph with $\mbox{ Mad}(G) < \frac{ 18}{ 7}$ and girth at least 6, then $\chi_s(G)\leq 5$. (3) If $G$ is a graph with $\mbox{ Mad}(G) < \frac{ 8}{ 3}$ and girth at least 6, then $\chi_s(G)\leq 6$. These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets.
Categories: math.CO