A computational approach to Conway's thrackle conjecture

Authors:
Radoslav Fulek;János Pach
Affiliations:
Ecole Polytechnique Fédérale de Lausanne, Switzerland;Ecole Polytechnique Fédérale de Lausanne, Switzerland and City College, New York, United States
Venue:
Computational Geometry: Theory and Applications
Year:
2011

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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 @e0, we give an algorithm terminating in e^O^(^(^1^/^@e^^^2^)^l^n^(^1^/^@e^)^) steps to decide whether t(n)==3. Using this approach, we improve the best known upper bound, t(n)=