The maximum number of digons formed by pairwise crossing pseudocircles

TL;DR

This work proves Grünbaum's conjecture that any simple arrangement of $n>2$ pairwise crossing pseudocircles has at most $2n-2$ digons, extending prior results from cylindrical and circular cases to all simple arrangements. The authors construct a digon graph and its bipartite double cover, representing each pseudocircle by two intersection points with a transversal curve $c$ and drawing edges along $c$ with carefully defined inside/outside passes. They show that every pair of independent edges crosses an even number of times, and apply the Strong Hanani–Tutte theorem to conclude planarity, yielding a tight bound of $4n-4$ edges in the double cover and thus at most $2n-2$ digons. The approach unifies earlier methods and provides a complete, self-contained proof, with further implications for touching points and potential extensions to other surfaces or triangle counts in pseudocircle arrangements.

Abstract

In 1972, Branko Grünbaum 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 Grünbaum's conjecture that has stood open for over five decades.

Figures (11)

• Figure 1: A family of 6 pseudocircles forming 10 digons.
• Figure 2: If we allow touchings and $n=3$, then a digon might be surrounded by only one pseudocircle.
• Figure 3: A lens and a lune.
• Figure 4: A pseudocircle cannot be both interior and exterior.
• Figure 5: Extending ${\cal{F}}$ with a curve $c$ which avoids the digons in ${\cal{A}}({\cal{F}})$ and intersects every pseudocircle in ${\cal{F}}$ exactly twice.

Theorems & Definitions (18)

• Conjecture 1: Grünbaum’s digon conjecture G72
• Theorem 1
• Corollary 1
• Proposition 1
• proof
• Lemma 1
• proof
• Proposition 2
• proof
• Theorem 2: Strong Hanani-Tutte Theorem Tutte70