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

University of Haifa, Technion - Israel Institute of Technology

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	American 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

Keywords

Circles
Digons
Grünbaum
Pseudo-circles

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics
Computational Mathematics

Access to Document

10.1007/s00493-025-00139-1

Cite this

@article{0c92292a06974cbab9089a37c674a287,

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.",

keywords = "Circles, Digons, Gr{\"u}nbaum, Pseudo-circles",

author = "Eyal Ackerman and G{\'a}bor Dam{\'a}sdi and Bal{\'a}zs Keszegh and Rom Pinchasi and Rebeka Raffay",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive licence to J{\'a}nos Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2025.",

year = "2025",

month = jun,

doi = "10.1007/s00493-025-00139-1",

language = "American English",

volume = "45",

journal = "Combinatorica",

issn = "0209-9683",

publisher = "Janos Bolyai Mathematical Society",

number = "3",

}

TY - JOUR

T1 - On the Number of Digons in Arrangements of Pairwise Intersecting Circles

AU - Ackerman, Eyal

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

Y1 - 2025/6

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.

AB - 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.

KW - Circles

KW - Digons

KW - Grünbaum

KW - Pseudo-circles

UR - http://www.scopus.com/inward/record.url?scp=105006429704&partnerID=8YFLogxK

U2 - 10.1007/s00493-025-00139-1

DO - 10.1007/s00493-025-00139-1

M3 - Article

SN - 0209-9683

VL - 45

JO - Combinatorica

JF - Combinatorica

IS - 3

M1 - 30

ER -

On the Number of Digons in Arrangements of Pairwise Intersecting Circles

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this