A thrackle on n vertices has fewer than 4n/3 edges

A thrackle is a drawing of a graph in the plane in which every two edges meet exactly once, either at a common endpoint or at a proper crossing. Conway conjectured that a thrackle with n vertices has at most n edges. We prove that a thrackle with n ≥ 4 vertices has at most 4(n − 1)/3 edges, improving the bound 18(n − 1)/13 obtained by Tóth and, independently, by Scoppetto. The proof combines Xu's discharging argument with two computer searches, by the method of Fulek and Pach, showing that two dumbbells are not thrackleable: a 6-cycle and an 8-cycle sharing one edge, and a 6-cycle and a 10-cycle sharing a path of three edges. A formal proof in Lean 4 derives the bound from five properties of thrackles and checks the computational facts it uses; Section 6 states what this proof trusts.

The programs, input graphs, filter lists, split states, logs and the Lean project are archived at doi:10.5281/zenodo.23716511 and available at github.com/tejasvi/thrackle-bound (tag v4, commit 3419264d42b3).