Please use this identifier to cite or link to this item: http://dx.doi.org/10.14279/depositonce-11610
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Main Title: Arrangements of pseudocircles: Triangles and drawings
Author(s): Felsner, Stefan
Scheucher, Manfred
Type: Article
URI: https://depositonce.tu-berlin.de/handle/11303/12810
http://dx.doi.org/10.14279/depositonce-11610
License: https://creativecommons.org/licenses/by/4.0/
Abstract: A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells p_3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least 2n-4. We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family of intersecting digon-free arrangements with p 3 ( A ) / n → 16 / 11 = 1 . 45 ¯ . We expect that the lower bound p 3 ( A ) ≥ 4 n / 3 is tight for infinitely many simple arrangements. It may however be true that all digon-free arrangements of  n pairwise intersecting circles have at least 2n-4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of p 3 ≥ 2 n / 3 , and conjecture that p 3 ≥ n - 1 . Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that p 3 ≤ 4/3 (n 2) + O ( n ) . This is essentially best possible because there are families of pairwise intersecting arrangements of n pseudocircles with p 3 = 4/3 (n 2) . The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by our generation algorithm. In the final section we describe some aspects of the drawing algorithm.
Subject(s): arrangement
circularizability
Grünbaum’s conjecture
pseudocircle
triangle
Tutte drawing
Issue Date: 27-Jan-2020
Date Available: 12-Mar-2021
Language Code: en
DDC Class: 510 Mathematik
Sponsor/Funder: TU Berlin, Open-Access-Mittel – 2020
Journal Title: Discrete and Computational Geometry
Publisher: SpringerNature
Volume: 65
Issue: 1
Publisher DOI: 10.1007/s00454-020-00173-4
Page Start: 261
Page End: 278
EISSN: 1432-0444
ISSN: 0179-5376
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik » FG Diskrete Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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