On odd and strong odd colorings of graphs

Abstract

An odd k-coloring of a graph G is a proper k-coloring such that for every non-isolated vertex v there is a color that occurs an odd number of times in the neighborhood of v. A strong odd k-coloring of G is a proper k-coloring such that for every vertex v every color occurs an odd number of times or 0 times in the neighborhood of v, which is a strengthened version of odd coloring and also a relaxation of square coloring. The odd chromatic number (or the strong odd chromatic number) of a graph G, denoted by ${\chi }_o\left(G\right)$ (or ${\chi }_{so}\left(G\right)$), is the minimum number of colors in any odd coloring (or strong odd coloring) of the graph G. In this paper, we prove that for any ${ϵ}_1>0$ and ${ϵ}_2\in (0,1)$, there exists a $\Delta ({ϵ}_1,{ϵ}_2)$ such that if $\Delta \left(G\right)\ge \Delta ({ϵ}_1,{ϵ}_2)$ and $\delta \left(G\right)\ge \Delta {\left(G\right)}^{\frac{1}{2}+{ϵ}_1}$, then ${\chi }_o\left(G\right)\le \chi \left(G\right)+\lceil {ϵ}_2\Delta \left(G\right)\rceil$, where $\chi \left(G\right)$ is the chromatic number of G, and $\Delta \left(G\right),\delta \left(G\right)$ are the maximum degree and minimum degree of G respectively. In addition, we construct a planar graph with strong odd chromatic number 13, which answers a question asked by Caro, Petruševski, Škrekovski and Tuza in negative.