Remarks on odd colorings of graphs

TL;DR

This paper explores the properties of odd colorings in graphs, introduces the odd chromatic number, and characterizes specific graph classes, providing bounds and raising open questions about this new coloring concept.

Contribution

It introduces and analyzes the odd chromatic number, characterizes certain graph classes, and establishes bounds related to maximum degree and degeneracy.

Findings

Acyclic graphs have specific odd chromatic numbers.

Hypercubes are characterized by their odd chromatic number.

Upper bounds are established based on maximum degree and degeneracy.

Abstract

A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of $G$, denoted $\chi_o(G)$, is the odd chromatic number. Odd colorings were recently introduced in [M.~Petru\v{s}evski, R.~\v{S}krekovski: \textit{Colorings with neighborhood parity condition}]. Here we discuss various basic properties of this new graph parameter, characterize acyclic graphs and hypercubes in terms of odd chromatic number, establish several upper bounds in regard to degenericity or maximum degree, and pose several questions and problems.