Tangled Thrackles

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. © 2012 Springer-Verlag.


Editor(s):
Márquez, Alberto
Ramos, Pedro
Urrutia, Jorge
Published in:
Computational Geometry, 7579, 45-53
Year:
2012
Publisher:
Berlin, Heidelberg, Springer Berlin Heidelberg
ISSN:
0925-7721
Laboratories:




 Record created 2014-07-28, last modified 2018-09-13


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