A computational approach to Conway's thrackle conjecture

  • Authors:

  • Radoslav Fulek;János Pach

  • Affiliations:

  • Ecole Polytechnique Fédérale de Lausanne;Ecole Polytechnique Fédérale de Lausanne and City College, New York

  • Venue:
  • GD'10 Proceedings of the 18th international conference on Graph drawing
  • Year:
  • 2010

Quantified Score

Hi-index 0.00

Visualization

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 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 167/117n n.