Arrangements of Pseudocircles: On Digons and Triangles

TL;DR

This paper investigates the maximum number of digons and minimum number of triangles in arrangements of pseudocircles, confirming longstanding conjectures and providing new constructions that clarify these combinatorial properties.

Contribution

The authors prove Gr"unbaum's conjecture on the maximum number of digons for arrangements with three touching pseudocircles and establish the exact minimum number of triangles in arrangements without digons or touchings.

Findings

Maximum of 2n - 2 touchings in arrangements with three touching pseudocircles.

Construction of arrangements with 2n - 2 touchings without triple touchings.

Confirmation that the minimum number of triangles is exactly rac{4}{3}n in certain arrangements.

Abstract

In this article, we study the cell-structure of simple arrangements of pairwise intersecting pseudocircles. The focus will be on two problems of Gr\"unbaum (1972). First, we discuss the maximum number of digons or touching points. Gr\"unbaum conjectured that there are at most $2n - 2$ digon cells or equivalently at most $2n - 2$ touchings. Agarwal et al. (2004) verified the conjecture for cylindrical arrangements. We show that the conjecture holds for any arrangement which contains three pairwise touching pseudocircles. The proof makes use of the result for cylindrical arrangements. Moreover, we construct non-cylindrical arrangements which attain the maximum of $2n - 2$ touchings and have no triple of pairwise touching pseudocircles. Second, we discuss the minimum number of triangular cells (triangles) in arrangements without digons and touchings. Gr\"unbaum conjectured that such arrangements have $2n - 4$ triangles. Snoeyink and Hershberger (1991) established a lower bound of $\lceil \frac{4}{3}n \rceil$. Felsner and Scheucher (2017) disproved the conjecture and constructed a family of arrangements with only $\lceil \frac{16}{11}n \rceil$ triangles. We provide a construction which shows that $\lceil \frac{4}{3}n \rceil$ is the correct value.