Math Conjectures

Conway's Thrackle Conjecture

OPENMajorConjectureProposed 1969 · Standard version

Canonical statement

Let a finite simple graph GG be drawn in the plane with vertices as distinct points and edges as simple arcs, with no edge through a nonincident vertex and no three edges meeting at an interior point. Suppose every pair of distinct edges meets exactly once, either at their common endpoint or in one proper crossing. Then
∣E(G)∣≤∣V(G)∣. |E(G)|\leq |V(G)|.
View source LaTeX
Let a finite simple graph \(G\) be drawn in the plane with
vertices as distinct points and edges as simple arcs, with no edge through
a nonincident vertex and no three edges meeting at an interior point.
Suppose every pair of distinct edges meets exactly once, either at their
common endpoint or in one proper crossing. Then
\[
  |E(G)|\leq |V(G)|.
\]
GRAPH-012:[Woodall1971Thrackle][CairnsNikolayevsky2000][FulekPach2019Thrackle]
Graph theoryGeometry and topology

Notes

A thrackle is a drawing of a finite simple graph in the plane in which every pair of distinct edges meets exactly once — either at a shared endpoint or in a single proper crossing — under the usual nondegeneracy assumptions on the arcs. Conway conjectured in 1969 that any graph admitting such a drawing satisfies ∣E(G)∣≤∣V(G)∣|E(G)|\leq|V(G)|. The problem entered the published literature through Woodall's early study [Woodall1971Thrackle].

The bound is classical for thrackles drawn with straight-line edges, and the conjecture has been established in several restricted drawing models and for special graph families; Cairns and Nikolayevsky, in particular, obtained bounds for generalized thrackles [CairnsNikolayevsky2000]. For arbitrary thrackles the number of edges is known to be at most a constant multiple of the number of vertices, with the constant improved by Fulek and Pach [FulekPach2019Thrackle], but every known universal bound has coefficient strictly larger than 11.

Bringing that coefficient down to the conjectured 11 is exactly what remains; the conjecture is still open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.