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 $φ$ with the property that for each non-isolated vertex $v\in V(G)$, there exists a color $c$ such that the cardinality of $φ^{-1}(c)\cap N(v)$ is odd. The concept of odd colorings is introduced by Petruševski and Škrekovski. 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 $2θ(G)-1$ and girth at least $6$ is odd $6θ(G)$-colorable, where $θ(G)$ is the thickness of $G$.