论文标题
环形图的奇数最多是9
The odd chromatic number of a toroidal graph is at most 9
论文作者
论文摘要
众所周知，每个平面图都是$ 4 $ - 色。环形图是可以嵌入圆环的图。事实证明，每个环形图都是$ 7 $ - 色。如果每个非分离顶点具有至少一种在其附近的奇数次数，则称为图形的适当着色称为\ emph {奇数}。承认图形$ g $的奇数颜色的最小颜色用$χ_{o}（g）$表示。在本文中，我们证明，如果$ g $是tortoidal，则$χ_{o} \ left（{g} \ right）\ le9 $;请注意，$ k_7 $是一个环形图，上限不少于$ 7 $。
It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has at least one color that appears an odd number of times in its neighborhood. The smallest number of colors that admits an odd coloring of a graph $ G $ is denoted by $χ_{o}(G)$. In this paper, we prove that if $G$ is tortoidal, then $χ_{o}\left({G}\right)\le9$; Note that $K_7$ is a toroidal graph, the upper bound is no less than $7$.
