On the Number of Digons in Arrangements of Pairwise Intersecting Circles

Eyal Ackerman; Gábor Damásdi; Balázs Keszegh; Rom Pinchasi; Rebeka Raffay

doi:10.1007/s00493-025-00139-1

On the Number of Digons in Arrangements of Pairwise Intersecting Circles

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

Department of Mathematics, Physics and Computer Science (Oranim Campus)

Research output: Contribution to journal › Article › peer-review

Abstract

A long-standing open conjecture of Branko Grünbaum from 1972 states that any simple arrangement of n pairwise intersecting pseudocircles in the plane can have at most 2n-2 digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles 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 pairwise intersecting pseudocircles in which there are three pseudocircles every pair of which create a digon. In this paper we prove this over 50-year-old conjecture of Grünbaum for any simple arrangement of pairwise intersecting circles in the plane.

Original language	English
Article number	30
Journal	Combinatorica
Volume	45
Issue number	3
DOIs	https://doi.org/10.1007/s00493-025-00139-1
State	Published - Jun 2025

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2025.

Keywords

Circles
Digons
Grünbaum
Pseudo-circles

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Computational Mathematics

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10.1007/s00493-025-00139-1

Cite this

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title = "On the Number of Digons in Arrangements of Pairwise Intersecting Circles",

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

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AU - Damásdi, Gábor

AU - Keszegh, Balázs

AU - Pinchasi, Rom

AU - Raffay, Rebeka

N1 - Publisher Copyright: © The Author(s), under exclusive licence to János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2025.

PY - 2025/6

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N2 - A long-standing open conjecture of Branko Grünbaum from 1972 states that any simple arrangement of n pairwise intersecting pseudocircles in the plane can have at most 2n-2 digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles 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 pairwise intersecting pseudocircles in which there are three pseudocircles every pair of which create a digon. In this paper we prove this over 50-year-old conjecture of Grünbaum for any simple arrangement of pairwise intersecting circles in the plane.

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KW - Circles

KW - Digons

KW - Grünbaum

KW - Pseudo-circles

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ER -

On the Number of Digons in Arrangements of Pairwise Intersecting Circles

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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