论文标题
交叉引理数字
Crossing lemma for the odd-crossing number
论文作者
论文摘要
如果可以在平面上绘制$ 1 $ - 平面图，以使每个边缘上最多都有一个。众所周知，$ 1 $ - 平面图最多具有4n-8美元的边缘。我们证明了以下奇数概括。如果可以在平面中绘制图形，以使每个边缘最多越过另一个边缘{\ em奇数次}，则称为1-odd-planar，最多具有$ 5N-9 $的边缘。结果，如果相邻的边缘交叉均匀数量，我们会改善奇数跨数字的交叉引理中的常数。我们还为$ K $ -ODD-Planar图的边数提供了上限。
A graph is $1$-planar, if it can be drawn in the plane such that there is at most one crossing on every edge. It is known, that $1$-planar graphs have at most $4n-8$ edges. We prove the following odd-even generalization. If a graph can be drawn in the plane such that every edge is crossed by at most one other edge {\em an odd number of times}, then it is called 1-odd-planar and it has at most $5n-9$ edges. As a consequence, we improve the constant in the Crossing Lemma for the odd-crossing number, if adjacent edges cross an even number of times. We also give upper bound for the number of edges of $k$-odd-planar graphs.
