On the number of digons in arrangements of pairwise intersecting circles

TL;DR

This paper proves Gr"unbaum's 50-year-old conjecture that any simple arrangement of n pairwise intersecting circles in the plane has at most 2n-2 digons, extending previous partial results to all such arrangements.

Contribution

It establishes the conjecture for all simple arrangements of pairwise intersecting circles, removing previous restrictions and confirming the conjecture in full generality.

Findings

Proves the conjecture for all arrangements of pairwise intersecting circles.

Extends previous partial results to the general case.

Confirms the maximum number of digons is 2n-2 for such arrangements.

Abstract

A long-standing open conjecture of Branko Gr\"unbaum 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\"unbaum'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\"unbaum for any simple arrangement of pairwise intersecting circles in the plane.