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 $\chi_{so}(G)$ 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 $\chi_{so}(G)\le \Delta(G)+4$, 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 $\chi_{so}(G)\le \Delta(G)+3$.
From: Hyemin Kwon [view email]
[v1]
Mon, 22 Jan 2024 02:20:56 UTC (93 KB)
[v2]
Tue, 23 Jan 2024 04:16:36 UTC (93 KB)
[v3]
Tue, 16 Jun 2026 01:28:06 UTC (85 KB)
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