Document Open Access Logo

On the Number of Digons in Arrangements of Pairwise Intersecting Circles

Authors Eyal Ackerman, Gábor Damásdi , Balázs Keszegh , Rom Pinchasi, Rebeka Raffay

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2024.3.pdf
  • Filesize: 0.73 MB
  • 14 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Arrangement of pseudocircles
  • Counting touchings
  • Counting digons
  • Grünbaum’s conjecture

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A long-standing open conjecture of Branko Grünbaum from 1972 states that any arrangement of n pairwise intersecting pseudocircles in the plane can have at most 2n-2 digons. Agarwal et al. proved this conjecture for arrangements in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Grünbaum’s conjecture is true for arrangements of pseudocircles in which there are three pseudocircles every pair of which creates a digon. In this paper we prove this over 50-year-old conjecture of Grünbaum for any arrangement of pairwise intersecting circles in the plane.

Cite As Get BibTex

Eyal Ackerman, Gábor Damásdi, Balázs Keszegh, Rom Pinchasi, and Rebeka Raffay. On the Number of Digons in Arrangements of Pairwise Intersecting Circles. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.3

Author Details

Eyal Ackerman
  • Department of Mathematics, Physics and Computer Science, University of Haifa at Oranim, Tivon 36006, Israel
Gábor Damásdi
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • ELTE Eötvös Loránd University, Budapest, Hungary
Balázs Keszegh
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • ELTE Eötvös Loránd University, Budapest, Hungary
Rom Pinchasi
  • Technion - Israel Institute of Technology, Haifa, Israel
  • Visiting professor at EPFL, Lausanne, Switzerland
Rebeka Raffay
  • École Polytechnique Fédérale de Lausanne, Switzerland

Funding

  • Damásdi, Gábor: Research partially supported by ERC grant No. 882971, "GeoScape".
  • Keszegh, Balázs: Research supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the National Research, Development and Innovation Office - NKFIH under the grant K 132696 and FK 132060, by the ÚNKP-23-5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund and by the ERC Advanced Grant "ERMiD". This research has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the ELTE TKP 2021-NKTA-62 funding scheme.
  • Pinchasi, Rom: Supported by ISF grant (grant No. 1091/21).

References

  1. Pankaj K. Agarwal, Eran Nevo, János Pach, Rom Pinchasi, Micha Sharir, and Shakhar Smorodinsky. Lenses in arrangements of pseudo-circles and their applications. J. ACM, 51(2):139-186, 2004. URL: https://doi.org/10.1145/972639.972641.
  2. Noga Alon, Hagit Last, Rom Pinchasi, and Micha Sharir. On the complexity of arrangements of circles in the plane. Discret. Comput. Geom., 26(4):465-492, 2001. URL: https://doi.org/10.1007/s00454-001-0043-x.
  3. Grant Cairns and Yury Nikolayevsky. Bounds for generalized thrackles. Discret. Comput. Geom., 23(2):191-206, 2000. URL: https://doi.org/10.1007/PL00009495.
  4. Grant Cairns and Yury Nikolayevsky. Generalized thrackle drawings of non-bipartite graphs. Discret. Comput. Geom., 41(1):119-134, 2009. URL: https://doi.org/10.1007/s00454-008-9095-5.
  5. Jordan S. Ellenberg, József Solymosi, and Joshua Zahl. New bounds on curve tangencies and orthogonalities. Discrete Anal., November 2016. URL: https://doi.org/10.19086/da.990.
  6. Paul Erdős. On sets of distances of n points. Am. Math. Mon., 53(5):248-250, 1946. URL: https://doi.org/10.2307/2305092.
  7. Stefan Felsner, Sandro Roch, and Manfred Scheucher. Arrangements of pseudocircles: on digons and triangles. In Graph drawing and network visualization, volume 13764 of Lecture Notes in Computer Science, pages 441-455. Springer, Cham, 2023. URL: https://doi.org/10.1007/978-3-031-22203-0_32.
  8. Branko Grünbaum. Arrangements and spreads. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 10. American Mathematical Society, Providence, R.I., 1972. Google Scholar
  9. Meir Katchalski and Hagit Last. On geometric graphs with no two edges in convex position. Discret. Comput. Geom., 19:399-404, 1998. URL: https://doi.org/10.1007/PL00009357.
  10. Adam Marcus and Gábor Tardos. Intersection reverse sequences and geometric applications. J. Comb. Theory, Ser. A, 113(4):675-691, 2006. URL: https://doi.org/10.1016/j.jcta.2005.07.002.
  11. János Pach and Pankaj K. Agarwal. Combinatorial Geometry, chapter 11, pages 177-178. John Wiley and Sons Ltd, 1995. URL: https://doi.org/10.1002/9781118033203.ch11.
  12. János Pach and Géza Tóth. Disjoint edges in topological graphs. In Combinatorial Geometry and Graph Theory, Indonesia-Japan Joint Conference, IJCCGGT 2003, Revised Selected Papers, volume 3330 of Lecture Notes in Computer Science, pages 133-140. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-30540-8_15.
  13. János Pach and Gábor Tardos. Forbidden paths and cycles in ordered graphs and matrices. Isr. J. Math., 155:359-380, 2006. URL: https://doi.org/10.1007/BF02773960.
  14. Rom Pinchasi. Gallai-Sylvester Theorem for Pairwise Intersecting Unit Circles. Discret. Comput. Geom., 28(4):607-624, 2002. URL: https://doi.org/10.1007/s00454-002-2892-3.
  15. Rom Pinchasi. A note on lenses in arrangements of pairwise intersecting circles in the plane, 2024. URL: https://arxiv.org/abs/2403.05270.
  16. Joel Spencer, Endre Szemerédi, and William T. Trotter. Unit distances in the Euclidean plane, pages 294-304. Academic Press, United Kingdom, 1984. Google Scholar
  17. László A. Székely. Crossing Numbers and Hard Erdős Problems in Discrete Geometry. Comb. Probab. Comput., 6(3):353-358, 1997. URL: https://doi.org/10.1017/S0963548397002976.
  18. William T. Tutte. Toward a theory of crossing numbers. J. Comb. Theory, 8(1):45-53, 1970. URL: https://doi.org/10.1016/S0021-9800(70)80007-2.
  19. Pavel Valtr. On geometric graphs with no k pairwise parallel edges. Discret. Comput. Geom., 19:461-469, 1998. URL: https://doi.org/10.1007/PL00009364.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact