论文标题
关于1平面图的奇数颜色的注释
A Note on Odd Colorings of 1-Planar Graphs
论文作者
论文摘要
如果每个未分离的顶点都有某种颜色在其附近出现奇怪的次数，则图形的适当着色是奇怪的。该概念最近是由Petruševski和škrekovski提出的，他们证明了每个平面图都承认了一个奇怪的$ 9 $颜色。他们还猜想，每个平面图都承认$ 5 $颜色。此后不久，克兰斯顿至少七个围墙的平面图证实了这一猜想。 Caro，Petruševski和škrekovski的外平面图。基于Caro，Petruševski和Škrekovski，Petr和Portier的工作，然后进一步证明，每个平面图都承认了一个奇怪的$ 8 $颜色。在本说明中，我们证明，每个1平面图都允许奇数$ 23 $颜色，如果可以将图形绘制在平面上，则图形为1平面，以便在最多另一个边缘越过每个边缘。
A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petruševski and Škrekovski, who proved that every planar graph admits an odd $9$-coloring; they also conjectured that every planar graph admits an odd $5$-coloring. Shortly after, this conjecture was confirmed for planar graphs of girth at least seven by Cranston; outerplanar graphs by Caro, Petruševski, and Škrekovski. Building on the work of Caro, Petruševski, and Škrekovski, Petr and Portier then further proved that every planar graph admits an odd $8$-coloring. In this note we prove that every 1-planar graph admits an odd $23$-coloring, where 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.
