On strong odd colorings of sparse graphs

A strong odd coloring of a graph is a proper vertex coloring such that for every vertex $v$, each color appearing on the neighborhood of $v$ appears an odd number of times. We investigate strong odd colorings of sparse graphs, where sparsity of a graph $G$ is measured in terms of maximum average degree, denoted $\operatorname{mad}(G)$. We prove that the optimal threshold on the maximum average degree guaranteeing a strong odd $c$-coloring is $4-Θ(\frac{1}{c})$ as $c\to\infty$. Moreover, for every $c\geq 5$, this threshold is strictly smaller than the threshold for odd $c$-coloring. We also determine the sharp threshold on the maximum average degree guaranteeing a strong odd $5$-coloring: every graph $G$ with $\operatorname{mad}(G)<\frac{7}{3}$ has a strong odd $5$-coloring, and this bound is sharp. In addition, we show that every graph $G$ with $\operatorname{mad}(G)<\frac{11}{5}$ and no induced $5$-cycle has a strong odd $4$-coloring. Note that the $5$-cycle has maximum average degree 2 and no strong odd $4$-coloring.