论文标题
每个环形图，没有$ 3 $ -CYCLES是奇怪的$ 7 $ - 可油
Every toroidal graph without $3$-cycles is odd $7$-colorable
论文作者
论文摘要
奇数着色是一种适当的着色，并具有额外的限制，即每个非分离顶点的颜色在其附近看起来有些奇数。 $ k $的最低颜色数量可以确保图$ g $的奇数颜色用$χ_o（g）$表示。我们说$ g $是$ k $ - 颜色，如果$χ_o（g）\ le K $。该概念是最近由Petruševski和škrekovski引入的，他们证明如果$ G $是平面，则$χ_{O}（g）（g）\ leq 9 $。环形图是可以嵌入圆环的图。请注意，$ k_7 $是环形图，$χ_{o}（g）\ leq7 $。在本文中，我们证明，没有$ 3 $ CYCLE的每个环形图都是奇怪的$ 7 $ - 颜色。因此，每个没有$ 3 $ CYCLES的平面图都是奇怪的$ 7 $ - 可油的推论。也就是说，如果剩余的案例解决了$ 3 $循环，则每个环形图都可以证明$ 7 $ - 可以证明。
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 $χ_o(G)$. We say $G$ is $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{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, $χ_{o}(G)\leq7$. 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.
