Star coloring of sparse graphs

TL;DR

This paper investigates the star chromatic number of sparse graphs, establishing upper bounds based on maximum average degree and girth, and introduces a decomposition approach into forests and independent sets.

Contribution

It provides new bounds on the star chromatic number for graphs with certain sparsity and girth conditions, using a novel decomposition method.

Findings

Graphs with Mad < 26/11 have star chromatic number at most 4.

Graphs with Mad < 18/7 and girth ≥ 6 have star chromatic number at most 5.

Graphs with Mad < 8/3 and girth ≥ 6 have star chromatic number at most 6.

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 $\chi_s(G)\leq 4$. (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.