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

An odd coloring of a graph G is a proper vertex coloring φ with the property that for each non-isolated vertex v∈ V(G), there exists a color c such that the cardinality of φ⁻¹(c)∩ N(v) is odd. The concept of odd colorings is introduced by Petru{s}evski and {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 2θ(G)-1 and girth at least $6$ is odd 6θ(G)-colorable, where θ(G) is the thickness of G.