Every toroidal graphs without adjacent triangles is odd 8-colorable

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.