| Title: | A note on the 2-colored rectilinear crossing number of random point sets in the unit square |
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| Authors: | ID Cabello, Sergio (Author)
ID Czabarka, Éva (Author)
ID Fabila-Monroy, Ruy (Author)
ID Higashikawa, Yuya (Author)
ID Seidel, Raimund (Author)
ID Székely, László (Author)
ID Tkadlec, Josef (Author)
ID Wesolek, Alexandra (Author) |
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| Files: |  PDF - Presentation file, download (504,15 KB)
MD5: B9EB5C7AC15B97E236601B03BBA82579
 URL - Source URL, visit https://link.springer.com/article/10.1007/s10474-024-01436-9
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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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. 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 ${1 \over 2} - {7 \over 50}$ of the total number of crossings. |
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| Keywords: | arrangement of points, flat, hyperplane |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.06.2024 |
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| Year of publishing: | 2024 |
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| Number of pages: | str. 214-226 |
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| Numbering: | Vol. 173, iss. 1 |
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| PID: | 20.500.12556/DiRROS-20508  |
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| UDC: | 514.17 |
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| ISSN on article: | 0236-5294 |
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| DOI: | 10.1007/s10474-024-01436-9 |
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| COBISS.SI-ID: | 206428931 |
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| Note: |
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| Publication date in DiRROS: | 03.10.2024 |
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| Views: | 621 |
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| Downloads: | 369 |
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