A thrackle on n vertices has fewer than 4n/3 edges
Tejasvi Singh Tomar · Zenodo (CERN European Organization for Nuclear Research) · 2026
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 51(n − 1)/38 edges, improving the bound 18(n − 1)/13 obtained by Tóth and, independently, by Scoppetto. The main new ingredient, suggested by Scoppetto, is that the dumbbell DB(6, 8, −1), a 6-cycle and an 8-cycle sharing one edge, is not thrackleable, which we show by a computer search. It follows, by an enumeration, that no 6-cycle of a triangle-free thrackleable graph shares an edge with an 8-cycle. With a second enumeration, which shows that a face of size 10 shares at most seven edges with faces of size 6, Xu's discharging argument then gives 51/38. The search tests the planarity of graphs that encode the orders of the crossings along the edges, following Fulek and Pach, and prunes with the lists of all such orders for subgraphs; for DB(6, 8, −1) it makes about 2.1·1011 planarity tests. We repeated the computations with a second implementation, written independently of the first. The programs, input graphs, filter lists, split states and logs of all computations are archived at doi:10.5281/zenodo.23549051 and available at github.com/tejasvi/thrackle-bound.