A computational approach to Conway's 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 50 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, t(n) ≤ 3 2 (n − 1), due to Cairns and Nikolayevsky, to 167

• Type: Journal
• Year: 2011
• Where: COMGEO
• Authors: Radoslav Fulek, János Pach
• Keywords: Applied Computing, COMGEO 2011, Common Vertex, Maximum Number, Thrackle