Every toroidal graphs without adjacent triangles is odd 8-colorable

TL;DR

This paper proves that all toroidal graphs lacking adjacent triangles can be colored with eight colors under the odd coloring rule, improving previous bounds for such graph classes.

Contribution

It establishes that toroidal graphs without adjacent triangles are odd 8-colorable, reducing the known upper bound from 9 to 8 colors.

Findings

All such graphs are odd 8-colorable.

Improves the upper bound from 9 to 8 colors.

Extends understanding of odd coloring in toroidal graphs.

Abstract

Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $\chi_o(G)$. We say $G$ is odd $k$-colorable if $\chi_o(G)\le k$. This notion is introduced very recently by Petru\v{s}evski and \v{S}krekovski, who proved that if $G$ is planar then $ \chi_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $\chi_{o}(G)\leq7$. Tian and Yin proved that every toroidal graph is odd $9$-colorable and every toroidal graph without $3$-cycles is odd $9$-colorable. In this paper, we proved that every toroidal graph without adjacent $3$-cycles is odd $8$-colorable.