Thrackles: An Improved Upper Bound

@inproceedings{Fulek2017ThracklesAI,
  title={Thrackles: An Improved Upper Bound},
  author={Radoslav Fulek and J{\'a}nos Pach},
  booktitle={Graph Drawing},
  year={2017}
}
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… 
View PDF on arXiv
3 Citations
Background Citations
1

3 Citations

Annular and pants thrackles
TLDR
The Thrackle Conjecture is proved for thrackle drawings all of whose vertices lie on the boundaries of connected domains in the complement of the drawing.
Great-circle Tree Thrackles
A thrackle is a graph drawing in which every pair of edges meets exactly once. Conway’s Thrackle Conjecture states that the number of edges of a thrackle cannot exceed the number of its vertices.…
A Note on the Maximum Rectilinear Crossing Number of Spiders
The maximum rectilinear crossing number of a graph $G$ is the maximum number of crossings in a good straight-line drawing of $G$ in the plane. In a good drawing any two edges intersect in at most one…

References

SHOWING 1-9 OF 9 REFERENCES
Bounds for Generalized Thrackles
TLDR
A bipartite graph G can be drawn as a generalized thrackle on a closed orientable connected surface if and only if G can been embedded in that surface.
On Conway's thrackle conjecture
TLDR
It is shown that a thrackle has at most twice as many edges as vertices as it was conjectured that it cannot exceed the number of its vertices.
On the Bounds of Conway’s Thrackles
A thrackle on a surfaceX is a graph of size e and order n drawn on X such that every two distinct edges of G meet exactly once either at their common endpoint, or at a proper crossing. An unsolved…
Generalized Thrackle Drawings of Non-bipartite Graphs
TLDR
It is shown that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1.
A computational approach to Conway's thrackle conjecture
Disjoint edges in topological graphs and the tangled-thrackle conjecture
Nikolayevsky. Bounds for generalized thrackles
  • Discrete Comput. Geom.,
  • 2000
An examination of the different possible solutions of a problem in incomplete blocks.
Thrackles and deadlock
  • Combinatorial Mathematics and Its Applications,
  • 1971