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 mad(G). We prove that the optimal threshold on the maximum average degree guaranteeing a strong odd c-coloring is 4-Θ(1/c) as c→∞. Moreover, for every c≥ 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 mad(G)<7/3 has a strong odd $5$-coloring, and this bound is sharp. In addition, we show that every graph G with mad(G)<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.