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<p>A long-standing open conjecture of Branko Grünbaum from 1972 states that any simple arrangement of <i>n</i> pairwise intersecting pseudocircles in the plane can have at most <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="493_2025_139_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(2n-2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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On the Number of Digons in Arrangements of Pairwise Intersecting Circles

  • 期刊论文

  • 发表日期：

    2025年5月26日

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

摘要

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 \(2n-2\) 2 n - 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.