论文标题
1平面图是奇数13色
1-planar graphs are odd 13-colorable
论文作者
论文摘要
图$ g $的奇怪着色是一种适当的着色，因此$ g $中的任何非分离顶点都在其邻居上都有奇怪的时光。用$χ_o（g）$表示的奇数色编号是最低颜色的数量，该颜色允许$ g $的奇数颜色。 Petruševski和Škrekovski在2021年提出了这一概念，并证明了如果$ g $是平面，则$χ_o（g）\ le9 $，并推测$χ_o（g）\ le5 $。最近，PETR和Portier将$ 9 $提高到$ 8 $。如果可以在平面中绘制图形，则图形为$ 1 $ - 平面，以便每个边缘在另一个边缘上都越过。克兰斯顿（Cranston），拉弗蒂（Lafferty）和宋（Song）表明，每$ 1 $ - 平面图都是奇怪的$ 23 $ - 色。在本文中，我们改进了这一结果，并表明每1美元的平面图都是奇怪的$ 13 $ - 色。
An odd coloring of a graph $G$ is a proper coloring such that any non-isolated vertex in $G$ has a coloring appears odd times on its neighbors. The odd chromatic number, denoted by $χ_o(G)$, is the minimum number of colors that admits an odd coloring of $G$. Petruševski and Škrekovski in 2021 introduced this notion and proved that if $G$ is planar, then $χ_o(G)\le9$ and conjectured that $χ_o(G)\le5$. More recently, Petr and Portier improved $9$ to $8$. A graph is $1$-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. Cranston, Lafferty and Song showed that every $1$-planar graph is odd $23$-colorable. In this paper, we improved this result and showed that every $1$-planar graph is odd $13$-colorable.
