The maximum number of digons formed by pairwise crossing pseudocircles

TL;DR

This paper proves Gr"unbaum's longstanding conjecture that any simple arrangement of pairwise crossing pseudocircles in the plane can have at most 2n-2 digons, extending previous results to all such arrangements.

Contribution

It provides the first complete proof of Gr"unbaum's conjecture for all simple arrangements of pairwise crossing pseudocircles, generalizing prior special case results.

Findings

Confirmed the maximum of 2n-2 digons for all simple arrangements

Extended previous results from cylindrical arrangements and circles

Established a complete proof of the conjecture

Abstract

In 1972, Branko Gr\"unbaum conjectured that any arrangement of $n>2$ pairwise crossing pseudocircles in the plane can have at most $2n-2$ digons (regions enclosed by exactly two pseudoarcs), with the bound being tight. While this conjecture has been confirmed for cylindrical arrangements of pseudocircles and more recently for geometric circles, we extend these results to any simple arrangement of pairwise intersecting pseudocircles. Using techniques from the above-mentioned special cases, we provide a complete proof of Gr\"unbaum's conjecture that has stood open for over five decades.