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Spanish Meeting on Computational Geometry

EGC 2011: Computational Geometry pp 45–53Cite as

Tangled Thrackles

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7579)

Abstract

A tangle is a graph drawn in the plane so that any pair of edges have precisely one point in common, and this point is either an endpoint or a point of tangency. If we allow a third option: the common point may be a proper crossing between the two edges, then the graph is called a tangled thrackle. We establish the following analogues of Conway’s thrackle conjecture: The number of edges of a tangle cannot exceed its number of vertices, n. We also prove that the number of edges of an x-monotone tangled thrackle with n vertices is at most n + 1. Both results are tight for n > 3. For not necessarily x-monotone tangled thrackles, we have a somewhat weaker, but nearly linear, upper bound.

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Author information

Authors and Affiliations

  1. Ecole Polytechnique Fédérale de Lausanne, Hungary

    János Pach

  2. Baruch College, City University of New York, USA

    Radoš Radoičić

  3. Rényi Institute of Mathematics, Budapest, Hungary

    Géza Tóth

Authors
  1. János Pach
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  2. Radoš Radoičić
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  3. Géza Tóth
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Editor information

Editors and Affiliations

  1. Escuela Técnica Superior de Ingenieria Informática, Universidad de Sevilla, Avenida Reina Mercedes, s/n, 41012, Sevilla, Spain

    Alberto Márquez

  2. Escuela Politécnica Superior, Universidad de Alcalá, E226, Campus Universitario, Alcalá de Henares, 28871, Madrid, Spain

    Pedro Ramos

  3. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Coyoacán, 04510, México. D.F, México

    Jorge Urrutia

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Pach, J., Radoičić, R., Tóth, G. (2012). Tangled Thrackles. In: Márquez, A., Ramos, P., Urrutia, J. (eds) Computational Geometry. EGC 2011. Lecture Notes in Computer Science, vol 7579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34191-5_4

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  • DOI: https://doi.org/10.1007/978-3-642-34191-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

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