Upper Bounds of the Odd Chromatic Number of a Graph in terms of its Thickness

Abstract

An odd coloring of a graph $G$ is a proper vertex coloring $\varphi$ with the property that for each non-isolated vertex $v\in V(G)$, there exists a color $c$ such that the cardinality of $\varphi^{-1}(c)\cap N(v)$ is odd. The concept of odd colorings is introduced by Petru\v{s}evski and \v{S}krekovski. In this paper, we investigate upper bounds of the odd chromatic number of a graph in terms of its thickness and other graphical parameters. In particular, we show that a graph $G$ with the minimum degree at least $2\theta(G)-1$ and girth at least $6$ is odd $6\theta(G)$-colorable, where $\theta(G)$ is the thickness of $G$.