The odd chromatic number of a toroidal graph is at most 9

TL;DR

This paper establishes that the odd chromatic number of any toroidal graph is at most 9, providing a new upper bound that advances understanding of coloring properties on toroidal surfaces.

Contribution

The paper proves a new upper bound of 9 for the odd chromatic number of toroidal graphs, improving previous bounds and contributing to graph coloring theory on surfaces.

Findings

The odd chromatic number of toroidal graphs is at most 9.

The bound is tight for the complete graph K7.

This result extends coloring theory to graphs on a torus.

Abstract

It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has at least one color that appears an odd number of times in its neighborhood. The smallest number of colors that admits an odd coloring of a graph $ G $ is denoted by $\chi_{o}(G)$. In this paper, we prove that if $G$ is tortoidal, then $\chi_{o}\left({G}\right)\le9$; Note that $K_7$ is a toroidal graph, the upper bound is no less than $7$.