A computational approach to Conwayʼs thrackle conjecture

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 eO((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, , due to Cairns and Nikolayevsky, to .