On the Bounds of Conway’s Thrackles
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A thrackle on a surface X is a graph of size e and order n drawn on X such that every two distinct edges of G meet exactly once either at their common endpoint, or at a proper crossing. An unsolved conjecture of Conway (1969) asserts that \(e\le n\) for every thrackle on a sphere. Until now, the best known bound is \(e\le 1.428n\). By using discharging rules we show that \(e\le 1.4n-1.4\).
KeywordsThrackle Generalized thrackle Crossing number
Mathematics Subject Classification05C10 05C62 68R10
We would like to thank the referees for their great suggestions which helped us a lot in improving the presentation, especially for their suggestions on reducing the upper bound from 1.4n to \(1.4(n-1)\).
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