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On the Number of Digons in Arrangements of Pairwise Intersecting Circles

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

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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).

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

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.

Subject Classification

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

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References

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