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. [1] 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 halved. 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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Acknowledgements
This work was carried out during Crossing Number Workshop 2022, Strobl, Austria. We thank the organizers and participants for providing a fruitful research environment.
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A.Wesolek was supported by the Vanier Canada Graduate Scholarships program.
J.Tkadlec was supported by Charles University projects UNCE/SCI/004 and PRIMUS/24/SCI/012.
Funded in part by the Slovenian Research and Innovation Agency (P1-0297, J1-2452, N1-0218, N1-0285). Funded in part by the European Union (ERC, KARST, project number 101071836). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
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Cabello, S., Czabarka, É., Fabila-Monroy, R. et al. A note on the 2-colored rectilinear crossing number of random point sets in the unit square. Acta Math. Hungar. 173, 214–226 (2024). https://doi.org/10.1007/s10474-024-01436-9
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DOI: https://doi.org/10.1007/s10474-024-01436-9