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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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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

  • Print ISBN: 978-3-642-34190-8

  • Online ISBN: 978-3-642-34191-5

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