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 several times. First, it was improved to 3(n − 1)/2, and another improvement led to a bound of roughly 1.428n. Moreover, the method used to prove the latter result yields for any ε > 0 a finite algorithm that either improves the bound to (1 + ε)n or disproves the conjecture. The current record is due to , who proved a bound of 1.3984n. 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. References External links thrackle.org—website about the problem Conjectures Topological graph theory Geometric intersection