Strong odd coloring of sparse graphs

Abstract

An odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, there is a color appearing an odd number of times in $N_G(v)$. Odd coloring of graphs was studied intensively in recent few years. In this paper, we introduce the notion of a strong odd coloring, as not only a strengthened version of odd coloring, but also a relaxation of square coloring. A strong odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, if a color appears in $N_G(v)$, then it appears an odd number of times in $N_G(v)$. We denote by $\ddot{I}{‡}_{so}(G)$ the smallest integer $k$ such that $G$ admits a strong odd coloring with $k$ colors. We prove that if $G$ is a graph with $mad(G)\le \frac{20}{7}$, then $\ddot{I}{‡}_{so}(G)\le \hat{I}”(G)+4$, and the bound is tight. We also prove that if $G$ is a $C_4$-free graph with $mad(G)\le \frac{30}{11}$, then $\ddot{I}{‡}_{so}(G)\le \hat{I}”(G)+3$.

20 pages, 12 figures