TejasviTomar
October 2026 · Preprint

A thrackle on n vertices has fewer than 51n/38 edges

Tejasvi Singh Tomar

In plain words

A thrackle is a drawing of points joined by curves in which every two curves meet exactly once, at a shared endpoint or by crossing. Conway conjectured that a thrackle never has more curves than points. In 2026 Tóth and, separately, Scoppetto proved it has fewer than 18/13 times as many. This paper lowers the factor to 51/38. A computer search, repeated by a second program written independently, shows a ring of six curves and a ring of eight sharing one curve cannot be drawn this way; with further computer checks, Xu's counting argument gives the bound.

Abstract

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.

Two closed curves on white, with no axes or labels. Left: a teal curve through six small yellow dots that loops around and crosses itself, with pointed tips at the dots at the top and on the right. Right: a larger magenta curve through eight yellow dots that winds in rounded, petal-like lobes and crosses itself many times.
Illustration of thrackle drawings of the 6-cycle (left, teal) and the 8-cycle (right, magenta). The vertices are the yellow dots, and every two edges meet exactly once, either at a common vertex or at a single crossing. Each drawing is built from one of the crossing systems that the paper's search lists in the repository: the 8 complete planar systems of the 6-cycle and the 2,544 of the 8-cycle, which give, for each edge, the order in which the edges that share no endpoint with it cross it. For such a system the augmented graph of Fulek and Pach (paper, Section 2) is planar; we embed it, place it by Tutte's method and round its paths into the curves shown, checking that every two curves still meet exactly once. The paper's computer search shows that no such drawing exists for a 6-cycle and an 8-cycle that share one edge, the dumbbell DB(6, 8, −1); with two further computer enumerations and a discharging argument, this gives at most 51(n − 1)/38 edges for a thrackle with n ≥ 4 vertices.