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arXiv · 2312.01935

A Note on the 2-Colored Rectilinear Crossing Number of Random Point Sets in the Unit Square

Sergio Cabello·Éva Czabarka·Ruy Fabila-Monroy·Yuya Higashikawa·Raimund Seidel·László Székely·Josef Tkadlec·Alexandra Wesolek
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Abstract

Let $S$ be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of $S$ with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that $S$ defines a pair of crossing edges of the same color is equal to $1/4$. This is connected to a recent result of Aichholzer et al. [GD 2019] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halfed. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation $\frac{1}{2}-\frac{7}{50}$ of the total number of crossings.

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Sergio Cabello, Éva Czabarka, Ruy Fabila-Monroy, Yuya Higashikawa, Raimund Seidel, László Székely, Josef Tkadlec, Alexandra Wesolek. 2023-12-04. A Note on the 2-Colored Rectilinear Crossing Number of Random Point Sets in the Unit Square. https://arxiv.org/abs/2312.01935

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