# P3088: Conway’s thrackle conjecture

- ID: `P3088`
- Reference: `conway-thrackle-conjecture`
- Page: https://theoremdb.org/statements/P3088
- Record maturity: Reviewed problem with recorded work

## Problem

If a finite simple graph with \(n\) vertices and \(m\) edges has a thrackle drawing in the plane, must \(m\le n\)?

### Context

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.

### Problem setup

- **Definition (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 (proper crossing).** A transverse intersection lying in the interiors of both edge arcs.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2) [3](#reference-3) [4](#reference-4)

## Work

### Evidence for the current status

**Claim 1 (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.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: 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.

The exact unresolved remainder is: 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.

A complete resolution must meet the following acceptance conditions:
- 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.

### Background and intake notes

- 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.
- The release review checked 4 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): 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. [1](#reference-1) [2](#reference-2) [3](#reference-3) [4](#reference-4)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked 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. 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](#reference-1) [2](#reference-2) [3](#reference-3) [4](#reference-4)

### Open directions

- **Route 2** (reported): 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `conway-thrackle-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://doi.org/10.48550/arXiv.2506.11808
   - 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
   - preprint; primary source; arXiv:2506.11808, checked 2026-08-01; checked 2026-08-01
   - Open copy: https://arxiv.org/abs/2506.11808
   - Source use: original_summary
   - Explicitly states that Conway’s planar conjecture remains open and records m ≤ 1.393n as the best current general bound.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1016/j.amc.2020.125573
   - journal_article; primary source; checked 2026-08-01
   - Source 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.
3. <a id="reference-3"></a>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 https://doi.org/10.1016/j.disc.2025.114840
   - journal_article; primary source; checked 2026-08-01
   - Open copy: https://arxiv.org/abs/2307.03252
   - Source 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.
4. <a id="reference-4"></a>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 https://doi.org/10.1016/j.comgeo.2011.02.001
   - journal_article; primary source; checked 2026-08-01
   - Open copy: https://arxiv.org/abs/1002.3904
   - Source 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.