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

01/18/2023

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by   Ruy Fabila-Monroy, et al.

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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 ≥ 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.[GD 2019].