Arrangements of Pseudocircles: On Digons and Triangles
DOI:
https://doi.org/10.57717/cgt.v3i1.67Abstract
In this article, we study the cell-structure of simple arrangements of pairwise intersecting pseudocircles. The focus will be on two problems of Grünbaum (1972).
First, we discuss the maximum number of digons or touching points. Grünbaum 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. Grünbaum conjectured that such arrangements have 2n - 4 triangles. Snoeyink and Hershberger (1991) established a lower bound of ⌈(4/3)n⌉. Felsner and Scheucher (2017) disproved the conjecture and constructed a family of arrangements with only ⌈(16/11)n⌉ triangles. We provide a construction which shows that ⌈(4/3)n⌉ is the correct value.
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Copyright (c) 2024 Stefan Felsner, Sandro Roch, Manfred Scheucher
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