Arrangements of Pseudocircles: On Digons and Triangles
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In this article, we study the cell-structure of simple arrangements of pairwise intersecting pseudocircles. The focus will be on two problems from Grünbaum's monograph from the 1970's. 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 pseudocircles that pairwise form a touching. 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. Felsner and Scheucher (2017) showed that there exist arrangements with only ⌈16/11n ⌉ triangles, which disproved a conjecture of Grünbaum. Here we provide a construction with only ⌈4/3n ⌉ triangles. A corresponding lower bound was obtained by Snoeyink and Hershberger (1991).
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