A Note on the $k$-colored Crossing Ratio of Dense Geometric Graphs

TL;DR

This paper demonstrates that dense geometric graphs can be colored with any number of colors to significantly reduce the proportion of crossing edge pairs of the same color, generalizing previous results for the case when k=2.

Contribution

It establishes a universal bound for the crossing ratio in k-colored dense geometric graphs, extending prior work from the case k=2 to all k ≥ 2.

Findings

Existence of a constant c>0 for all k≥2 ensuring reduced same-color crossings.

Generalization of previous results from k=2 to arbitrary k in dense geometric graphs.

Quantitative bounds on crossing ratios in k-colored 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].