Crossing lemma for the odd-crossing number

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications Pub Date : 2023-01-01 DOI:10.1016/j.comgeo.2022.101901

János Karl , Géza Tóth

引用次数: 0

Abstract

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 $4 n - 8$ 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 an odd number of times, then it is called 1-odd-planar and it has at most $5 n - 9$ 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.

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奇数交叉数的交叉引理

如果图可以在平面中绘制，使得每条边上最多有一个交点，那么它就是1-平面的。众所周知，1-平面图最多有4n-8条边。我们证明了以下奇偶推广。如果一个图可以在平面上绘制，使得每条边最多与另一条边相交奇数次，那么它被称为1-odd-planar，并且最多有5n-9条边。因此，如果相邻边交叉偶数次，我们改进了奇数交叉数的交叉引理中的常数。给出了k-奇平面图的边数的上界。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Computational Geometry-Theory and Applications 数学-数学

CiteScore

1.60

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.