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Conway Thrackle Conjecture

The Conway thrackle conjecture asserts that every thrackle with n vertices and m edges satisfies m<=n.

Scoppetto (2026) proved the improved upper bound

m<=(18)/(13)(n-1)

for n>=4, using a computer-assisted exclusion of a particular drawing obstruction. The conjectured bound m<=n remains open.

Scoppetto (2026) reports using ChatGPT for algorithmic improvements, code implementation, proof checking, and identification of omitted cases. The author reviewed the computation and takes responsibility for the proof.

Simon (2026) further improved the upper bound to

m<=|_(4(n-1))/3_|

for every n>=4, where |_x_| is the floor function. The proof combines a cited computer-assisted verification for at most 12 vertices with a discharging argument (Simon 2026).

See also

Graph Crossing Number, Graph Drawing, Thrackle

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References

Scoppetto, T. "Another Brick in the Wall: A Computationally Assisted Proof of an Improved Upper Bound for Thrackles." 13 Oct 2026. https://arxiv.org/abs/2610.17508.Simon, D. G. "Conway's Thrackle Conjecture: A New Upper Bound of 4n/3." 22 Oct 2026. https://arxiv.org/abs/2610.31589.

Cite this as:

Weisstein, Eric W. "Conway Thrackle Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConwayThrackleConjecture.html

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