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On the number of digons in arrangements of pairwise intersecting circles

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

TL;DR

This work resolves Grünbaum's long-standing conjecture by proving that any simple arrangement of $n>2$ pairwise intersecting circles in the plane contains at most $2n-2$ digons, with the bound shown to be tight. The authors leverage three geometric lemmata about the centers' graph, distinguishing lune- and lens-edges, to control how digons can occur, and they employ an inversion-based simplification under the simplicity assumption. The key methodological step is a sphere-projection doubling trick that yields a planar bipartite graph $G'$ on $2n$ vertices with $2E$ edges, allowing the standard planar edge bound to force $E \\leq 2n-2$, hence the digon bound. The result extends previous partial cases and reinforces the special geometric properties of circles in digon configurations, contributing a definitive answer to Grünbaum's conjecture for simple circle arrangements.

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.

On the number of digons in arrangements of pairwise intersecting circles

TL;DR

This work resolves Grünbaum's long-standing conjecture by proving that any simple arrangement of pairwise intersecting circles in the plane contains at most digons, with the bound shown to be tight. The authors leverage three geometric lemmata about the centers' graph, distinguishing lune- and lens-edges, to control how digons can occur, and they employ an inversion-based simplification under the simplicity assumption. The key methodological step is a sphere-projection doubling trick that yields a planar bipartite graph on vertices with edges, allowing the standard planar edge bound to force , hence the digon bound. The result extends previous partial cases and reinforces the special geometric properties of circles in digon configurations, contributing a definitive answer to Grünbaum's conjecture for simple circle arrangements.

Abstract

A long-standing open conjecture of Branko Grünbaum from 1972 states that any simple arrangement of pairwise intersecting pseudocircles in the plane can have at most 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.
Paper Structure (3 sections, 6 theorems, 12 figures)

This paper contains 3 sections, 6 theorems, 12 figures.

Table of Contents

  1. Introduction
  2. Three crucial geometric lemmata
  3. Proof of Theorem \ref{['theorem:main']}

Key Result

Theorem 1

Every non-trivial simple arrangement of $n$ pairwise intersecting circles has at most $2n-2$ digons.

Figures (12)

  • Figure 1: A lens and a lune.
  • Figure 2: A family of $5$ pairwise intersecting circles with $8$ lenses.
  • Figure 3: A pair of avoiding edges.
  • Figure 4: An illustration for Observation \ref{['observation:simple']}.
  • Figure 5: An illustration for Observation \ref{['observation:d']}.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Conjecture 1: Grünbaum’s digon conjecture GRUNB72
  • Theorem 1
  • Theorem 2
  • Theorem 3: P24
  • Lemma 1: ALPS01
  • Lemma 2: P24
  • Lemma 3