A note on the k-colored crossing ratio of dense geometric graphs

IF 0.4 4区计算机科学 Q4 MATHEMATICS Computational Geometry-Theory and Applications Pub Date : 2024-07-22 DOI:10.1016/j.comgeo.2024.102123

Ruy Fabila-Monroy

引用次数: 0

Abstract

A geometric graph is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer $k \geq 2$ , there exists a constant $c > 0$ such that the following holds. The edges of every dense geometric graph, with sufficiently many vertices, can be colored with k colors, such that the number of pairs of edges of the same color that cross is at most $(1 / k - c)$ times the total number of pairs of edges that cross. The case when $k = 2$ and G is a complete geometric graph, was proved by Aichholzer et al. (2019) [2].

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关于密集几何图的 k 色交叉比的说明

几何图形是一种顶点集是平面上一般位置点的集合，边是连接这些点的直线段的图形。我们证明，对于每一个整数 k≥2，都存在一个常数 c>0，使得以下条件成立。每个具有足够多顶点的密集几何图形的边都可以用 k 种颜色着色，这样，交叉的同色边对数最多是交叉边对总数的（1/k-c）倍。Aichholzer 等人（2019）[2] 证明了 k=2 且 G 是完整几何图形时的情况。

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.

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