A Note on the $k$-colored Crossing Ratio of Dense Geometric Graphs
Abstract
A \emph{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 $k \ge 2$, there exists a constat $c>0$ such that the following holds. The edges of every dense geometric graph 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.[\emph{GD} 2019].
- Publication:
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arXiv e-prints
- Pub Date:
- January 2023
- DOI:
- 10.48550/arXiv.2301.07261
- arXiv:
- arXiv:2301.07261
- Bibcode:
- 2023arXiv230107261F
- Keywords:
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- Computer Science - Computational Geometry;
- Mathematics - Combinatorics