Intersecting Diametral Balls Induced by a Geometric Graph

Embargo End Date
2023-11-14

Type
Article

Authors
Pirahmad, Olimjoni
Polyanskii, Alexandr
Vasilevskii, Alexey

KAUST Department
Visual Computing Center (VCC)

Date
2022-11-14

Abstract
For a graph whose vertex set is a finite set of points in the Euclidean d-space consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect. Using the idea of halving lines, we show that (i) for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph; (ii) for any n red and n blue points in the plane, there exists a perfect red-blue matching that is a Tverberg graph. Also, we prove that (iii) for any even set of points in the Euclidean d-space, there exists a perfect matching that is an open Tverberg graph; (iv) for any n red and n blue points in the Euclidean d-space, there exists a perfect red-blue matching that is a Tverberg graph.

Citation
Pirahmad, O., Polyanskii, A., & Vasilevskii, A. (2022). Intersecting Diametral Balls Induced by a Geometric Graph. Discrete & Computational Geometry. https://doi.org/10.1007/s00454-022-00457-x

Acknowledgements
The research of the second and third authors was funded by the grant of Russian Science Federation No. 21-71-10092, https://rscf.ru/project/21-71-10092/ . The second author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury. The work was done when the first and third authors were students at MIPT. We thank Pablo Soberón for sharing the problem. Besides, we thank János Pach for drawing our attention to [9]. We are also grateful to the members of the Laboratory of Combinatorial and Geometric Structures at MIPT for the stimulating and fruitful discussions. We thanks to the referees for their suggestions that helped us to significantly improve the presentation of the paper.

Publisher
Springer Science and Business Media LLC

Journal
Discrete and Computational Geometry

DOI
10.1007/s00454-022-00457-x

arXiv
2108.09795

Additional Links
https://link.springer.com/10.1007/s00454-022-00457-x

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