A computational approach to Conway's thrackle conjecture

Abstract

A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 years old conjecture of Conway, t(n) = n for every n ≥ 3. For any ε > 0, we give an algorithm terminating in e O((1/ε2) ln(1/ε)) steps to decide whether t(n) ≤ (1 + ε)n for all n ≥ 3. Using this approach, we improve the best known upper bound, t(n) ≤ 3/2(n - 1), due to Cairns and Nikolayevsky, to 167/117 n < 1.428n. © 2011 Springer-Verlag.