Please use this identifier to cite or link to this item: http://dx.doi.org/10.14279/depositonce-11610
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dc.contributor.authorFelsner, Stefan-
dc.contributor.authorScheucher, Manfred-
dc.date.accessioned2021-03-12T07:38:43Z-
dc.date.available2021-03-12T07:38:43Z-
dc.date.issued2020-01-27-
dc.identifier.issn0179-5376-
dc.identifier.urihttps://depositonce.tu-berlin.de/handle/11303/12810-
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-11610-
dc.description.abstractA 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.en
dc.description.sponsorshipTU Berlin, Open-Access-Mittel – 2020en
dc.language.isoen-
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/-
dc.subject.ddc510 Mathematiken
dc.subject.otherarrangementen
dc.subject.othercircularizabilityen
dc.subject.otherGrünbaum’s conjectureen
dc.subject.otherpseudocircleen
dc.subject.othertriangleen
dc.subject.otherTutte drawingen
dc.titleArrangements of pseudocircles: Triangles and drawingsen
dc.typeArticleen
tub.accessrights.dnbfreeen
tub.publisher.universityorinstitutionTechnische Universität Berlinen
dc.identifier.eissn1432-0444-
dc.type.versionpublishedVersionen
dcterms.bibliographicCitation.doi10.1007/s00454-020-00173-4en
dcterms.bibliographicCitation.journaltitleDiscrete and Computational Geometryen
dcterms.bibliographicCitation.originalpublisherplaceLondon [u.a.]en
dcterms.bibliographicCitation.volume65en
dcterms.bibliographicCitation.pageend278en
dcterms.bibliographicCitation.pagestart261en
dcterms.bibliographicCitation.originalpublishernameSpringerNatureen
dcterms.bibliographicCitation.issue1en
tub.affiliationFak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik » FG Diskrete Mathematikde
Appears in Collections:Technische Universität Berlin » Publications

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