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 $χ_{so}(G)$ 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 $χ_{so}(G)\le Δ(G)+4$, and the bound is tight. We also prove that if $G$ is a graph with $mad(G)\le\frac{30}{11}$ and $Δ(G)\ge 4$, then $χ_{so}(G)\le Δ(G)+3$.

Figures (12)

• Figure 1: A Petersen graph $G$ with $\chi_{so}(G)=6$.
• Figure 2: A planar subcubic graph $G$ with $\chi(G^2)=\chi_{so}(G)=7$
• Figure 3: Illustrations of Lemma \ref{['lem:D+c']}.
• Figure 4: An illustration of Lemma \ref{['lem:D+4']}.
• Figure 5: Illustrations of Lemma \ref{['lem:vertexdistinct']}.

Theorems & Definitions (42)

• Conjecture 1.1: wegner1977
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3