Every toroidal graph without $3$-cycles is odd $7$-colorable

TL;DR

This paper proves that all toroidal graphs without 3-cycles can be properly colored with 7 colors under the odd coloring rule, extending known results for planar graphs and contributing to the understanding of odd colorings on surfaces.

Contribution

It establishes that toroidal graphs without 3-cycles are odd 7-colorable, a significant extension of previous planar graph results to toroidal graphs.

Findings

All toroidal graphs without 3-cycles are odd 7-colorable.

This result implies planar graphs without 3-cycles are also odd 7-colorable.

The work advances the theory of odd colorings on different surface embeddings.

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 $k$-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, $\chi_{o}(G)\leq7$. In this paper, we proved that, every toroidal graph without $3$-cycles is odd $7$-colorable. Thus, every planar graph without $3$-cycles is odd $7$-colorable holds as a corollary. That's to say, every toroidal graph is $7$-colorable can be proved if the remained cases around $3$-cycle is resolved.