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The Maximum Number of Digons Formed by Pairwise Intersecting Pseudocircles

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

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • pairwise intersecting arrangement
  • arrangement of pseudocircles
  • counting digons
  • tangencies
  • Grünbaum’s conjecture

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Abstract

In 1972, Branko Grünbaum conjectured that any nontrivial arrangement of n > 2 pairwise intersecting pseudocircles in the plane can have at most 2n-2 digons (regions enclosed by exactly two pseudoarcs), with the bound being tight. While this conjecture has been confirmed for cylindrical arrangements of pseudocircles and more recently for geometric circles, we extend these results to any simple arrangement of pairwise intersecting pseudocircles.

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Eyal Ackerman, Gábor Damásdi, Balázs Keszegh, Rom Pinchasi, and Rebeka Raffay. The Maximum Number of Digons Formed by Pairwise Intersecting Pseudocircles. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.2

Author Details

Eyal Ackerman
  • Department of Mathematics, Physics and Computer Science, University of Haifa at Oranim, Tivon, 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
Rebeka Raffay
  • École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

Funding

  • Damásdi, Gábor: Research supported by ERC grant No. 882971, "GeoScape" and by the Eköp-24 university excellence scholarship program of the Ministry for Culture and Innovation from the source of the National Research, Development and Innovation Fund.
  • 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.

References

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