[#P3088] Conway’s thrackle conjecture
Problem. If a finite simple graph with \(n\) vertices and \(m\) edges has a thrackle drawing in the plane, must \(m\le n\)?
1Context
Known frontier: Every planar thrackle satisfies m ≤ 1.393(n-1). The conjectured coefficient 1 is proved for straight-line thrackles and several other restricted drawing classes. Open boundary: Reduce the general coefficient to 1, or construct a planar thrackle with more edges than vertices. A recent higher-genus counterexample concerns a different conjecture and explicitly leaves the planar target open. A repository corpus search for thrackle returned no duplicate target.
2Problem setup
Definition 1 (thrackle drawing). Vertices are distinct points and edges are simple Jordan arcs such that adjacent edges meet only at their common endpoint and nonadjacent edges cross exactly once.
Definition 2 (proper crossing). A transverse intersection lying in the interiors of both edge arcs.
Remark 1. In a thrackle drawing, every pair of edges meets exactly once. Adjacent edges meet at their common endpoint, and nonadjacent edges cross once in their interiors. Odd cycles attain m=n. The question is whether a thrackle can contain more edges than vertices.
3What counts as a solution
- Prove m ≤ n for every finite simple graph admitting a planar thrackle drawing.
- Or give an explicit finite simple graph with m>n and a fully specified planar drawing, together with a rigorous check that every pair of edges meets exactly once in the required manner.
1Status
Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every planar thrackle satisfies m ≤ 1.393(n-1). The conjectured coefficient 1 is proved for straight-line thrackles and several other restricted drawing classes. Exact unresolved remainder: Reduce the general coefficient to 1, or construct a planar thrackle with more edges than vertices. A recent higher-genus counterexample concerns a different conjecture and explicitly leaves the planar target open. TheoremDB corpus searches for thrackle returned no duplicate target.[1][2][3][4]
1Records
Notes and companion material
Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Every planar thrackle satisfies m ≤ 1.393(n-1). The conjectured coefficient 1 is proved for straight-line thrackles and several other restricted drawing classes. Exact unresolved remainder: Reduce the general coefficient to 1, or construct a planar thrackle with more edges than vertices. A recent higher-genus counterexample concerns a different conjecture and explicitly leaves the planar target open. TheoremDB corpus searches for thrackle returned no duplicate target.
- Equivalent-formulation queries: Conway thrackle conjecture open 2026; Conway thrackle best known bound 1.393; thrackle straight line conjecture theorem
- Strongest checked neighboring result: Every planar thrackle satisfies m ≤ 1.393(n-1). The conjectured coefficient 1 is proved for straight-line thrackles and several other restricted drawing classes.
- Exact unresolved remainder: Reduce the general coefficient to 1, or construct a planar thrackle with more edges than vertices. A recent higher-genus counterexample concerns a different conjecture and explicitly leaves the planar target open. TheoremDB corpus searches for thrackle returned no duplicate target.
How the 4 records connect
ProblemConway’s thrackle conjecture
2See also
- Cycle Double Cover Conjecturecombinatorics
- The Total Coloring Conjecturecombinatorics
- Sabidussi's Compatibility Conjecturecombinatorics
How to cite
TheoremDB contributors, “Conway’s thrackle conjecture,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/conway-thrackle-conjectureThis page as plain text: conway-thrackle-conjecture.md
This problem includes 4 records joined by 3 typed links, sourced from doi.org[1], current as of August 1, 2026.
1References
- Packet source. C. Hernández-Vélez, J. Kynčl, and G. Salazar, Thrackles on nonplanar surfaces, arXiv:2506.11808, version dated March 22, 2026. Abstract and Introduction, especially the current planar status paragraph. ↗ open copy ↗preprint · primary source · arXiv:2506.11808, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Explicitly states that Conway’s planar conjecture remains open and records m ≤ 1.393n as the best current general bound.Also cited at C. Hernández-Vélez, J. Kynčl, and G. Salazar, Thrackles on nonplanar surfaces, arXiv:2506.11808, version dated March 22, 2026. Abstract and Introduction, especially the current planar status paragraph.Source used to assess the problem's recorded status.For Conway’s thrackle conjecture: This is the dated publication status for the canonical target Conway’s thrackle conjecture.Source named by the research packet.
- Yian Xu, “A New Upper Bound for Conway’s Thrackles”. Applied Mathematics and Computation 389 (2021), 125573. DOI 10.1016/j.amc.2020.125573. Abstract and main upper-bound theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves m ≤ 1.393(n-1) for simple connected thrackable graphs.Source used to assess the problem's recorded status.For Conway’s thrackle conjecture: Proves m ≤ 1.393(n-1) for simple connected thrackable graphs.
- Balázs Keszegh and Dániel Simon, “Convex hull thrackles”. Discrete Mathematics 349(3) (2026), 114840. DOI 10.1016/j.disc.2025.114840. Abstract and opening discussion of linear thrackles. ↗ open copy ↗journal article · primary source · checked 2026-08-01Source use: original summary.Records the planar conjecture and the known exact result for straight-line thrackles while studying a convex-hull extension.Source used to assess the problem's recorded status.For Conway’s thrackle conjecture: Records the planar conjecture and the known exact result for straight-line thrackles while studying a convex-hull extension.
- Radoslav Fulek and János Pach, “A computational approach to Conwayʼs thrackle conjecture”. Computational Geometry 44(6-7) (2011), 345-355. DOI 10.1016/j.comgeo.2011.02.001. Abstract and main algorithmic upper-bound result. ↗ open copy ↗journal article · primary source · checked 2026-08-01Source use: original summary.Gives a finite procedure for testing uniform bounds of the form t(n)<(1+ε)n and established the earlier 167n/117 bound.Source used to assess the problem's recorded status.For Conway’s thrackle conjecture: Gives a finite procedure for testing uniform bounds of the form t(n)<(1+ε)n and established the earlier 167n/117 bound.
Original TheoremDB editorial statement and source synthesis; external works are used for citation only.