Lecture Notes in Computer Science Volume 6502, 2011, pp 226-237

A Computational Approach to Conway’s Thrackle Conjecture

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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 ε> 0, we give an algorithm terminating in \(e^{O((1/\varepsilon^2)\ln(1/\varepsilon))}\) steps to decide whether t(n) ≤ (1 + ε)n for all n ≥ 3. Using this approach, we improve the best known upper bound, \(t(n)\le \frac 32(n-1)\) , due to Cairns and Nikolayevsky, to \(\frac{167}{117}n<1.428n\) .

Research partially supported by NSF grant CCF-08-30272, grants from OTKA, SNF, and PSC-CUNY.