Thrackles: An Improved Upper Bound

  title={Thrackles: An Improved Upper Bound},
  author={Radoslav Fulek and J{\'a}nos Pach},
  booktitle={Graph Drawing},
A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is \({3\over 2}(n-1)\), and that this bound is best… 
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Nikolayevsky. Bounds for generalized thrackles
  • Discrete Comput. Geom.,
  • 2000
Thrackles and deadlock
  • Combinatorial Mathematics and Its Applications,
  • 1971