PreprintCC BY-SA 4.0arXiv
Conway's thrackle conjecture: a new upper bound of 4/3n
Abstract
A thrackle is a drawing of a graph in the plane in which every pair of edges meets exactly once, either at a common endpoint or at a proper crossing. Conway conjectured that every thrackle has at most as many edges as vertices. We prove that every thrackle on n≥4 vertices has at most ⌊4(n−1)/3⌋ edges. The proof combines Pammer's computer-assisted verification of the conjecture for graphs on at most twelve vertices with a discharging argument in suitable plane and projective-plane embeddings.
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BibTeX
@misc{simon2026conways,
title = {{Conway's thrackle conjecture: a new upper bound of $\frac43n$}},
author = {Dániel Gábor Simon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31589},
eprint = {2610.31589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31589},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Conway's thrackle conjecture: a new upper bound of 4/3n
AU - Dániel Gábor Simon
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31589
UR - https://arxiv.org/abs/2610.31589
AB - A thrackle is a drawing of a graph in the plane in which every pair of edges meets exactly once, either at a common endpoint or at a proper crossing. Conway conjectured that every thrackle has at most as many edges as vertices. We prove that every thrackle on n≥4 vertices has at most ⌊4(n−1)/3⌋ edges. The proof combines Pammer's computer-assisted verification of the conjecture for graphs on at most twelve vertices with a discharging argument in suitable plane and projective-plane embeddings.
ER -