Conway’s Thrackle Conjecture

Prove or disprove Conway’s thrackle conjecture: Establish that every graph that admits a thrackle drawing—namely, a drawing in which each pair of non-adjacent edges cross exactly once and adjacent edges do not cross—has at most as many edges as vertices.

Background

The paper surveys the notion of thrackles, drawings achieving the thrackle bound where every non-adjacent pair of edges crosses exactly once. While some graph classes (e.g., cycles with more than four vertices and trees) admit thrackles, it is unknown in general which graphs do, and Conway’s long-standing conjecture asserts a tight edge bound for any thrackleable graph.

The authors reference the conjecture as part of the background motivating their work on drawings with prescribed crossings and controlled curve complexity.

References

As reported in, Conway conjectured in 1969 that each thrackleable graph contains at most as many edges as vertices. His conjecture, known as Conway's thrackle conjecture, still remains open.

— Tangling and Untangling Trees on Point-sets (2508.18535 - Battista et al., 25 Aug 2025) in Section 1, Introduction