Every toroidal graph without $3$-cycles is odd $7$-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 $k$-colorable if ${\chi }_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then ${\chi }_o(G)\le 9$. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, ${\chi }_o(G)\le 7$. 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.

arXiv comment9 pages