EGC 2011: Computational Geometry pp 45-53

# Tangled Thrackles

• János Pach
• Géza Tóth
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, 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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• János Pach
• 1