Known Bounds
Lovász et al. proved that every bipartite thrackle is a planar graph, although not drawn in a planar way. As a consequence, they show that every thrackleable graph with n vertices has at most 2n − 3 edges. Since then, this bound has been improved two times. First, it was improved to 3(n − 1)/2, and the current best bound is roughly 1.428n. Moreover, the method used to prove this result yields for any ε>0 a finite algorithm that either improves the bound to (1 + ε)n or disproves the conjecture.
If the conjecture is false, a minimal counterexample would have the form of two even cycles sharing a vertex. Therefore, to prove the conjecture, it would suffice to prove that graphs of this type cannot be drawn as thrackles.
Read more about this topic: Conway's Thrackle Conjecture
Famous quotes containing the word bounds:
“At bounds of boundless void.”
—Samuel Beckett (1906–1989)
“How far men go for the material of their houses! The inhabitants of the most civilized cities, in all ages, send into far, primitive forests, beyond the bounds of their civilization, where the moose and bear and savage dwell, for their pine boards for ordinary use. And, on the other hand, the savage soon receives from cities iron arrow-points, hatchets, and guns, to point his savageness with.”
—Henry David Thoreau (1817–1862)