Another Brick in the Wall: A Computationally Assisted Proof of an Improved Upper Bound for Thrackles

Let $G$ be a thrackle, where a thrackle is a graph drawing in which each edge meets every other edge exactly once, either at a vertex or at a crossing. It is conjectured that, for all thrackles, the number of edges is less than or equal to the number of vertices. Fulek and Pach created a program to determine whether or not a graph can be a thrackle. Through a reconstruction and improvement of their algorithm, we computationally prove a key dumbbell to be non-thrackleable. Using this exclusion, we improve Xu’s 8-face half-edge bound from five to four. Finally, using this stronger restriction with Xu’s framework, together with separate arguments for graphs containing triangles and disconnected graphs, we improve the known upper bound toward this conjecture to $m\le 1.385(n-1)$ for $n\ge 4$.