Conway’s thrackle conjecture

Can a finite drawing in the plane obey a one-meeting rule yet have more connections than dots? Use no connection from a dot to itself and at most one between each pair. Connections sharing a dot must meet only there; connections with no shared dot must cross exactly once. A triangle is a legal example with three dots and three connections: each pair meets at its shared corner. The conjecture says the number of connections never exceeds the number of dots, even when connections are curved. Crossings do not count as extra dots.[1]

Can a finite graph drawn in the plane with exactly one permitted meeting per edge pair have more edges than vertices? Such a drawing is called a thrackle. For a simple graph, there are no loops or repeated edges; adjacent edges meet only at their shared vertex, and nonadjacent edges cross exactly once. Draw five points around a convex pentagon and connect each to the point two steps ahead: the resulting five-point star has five edges and five vertices. Each pair with disjoint endpoints crosses once, so equality is possible. Conway predicts $|E|\le|V|$ for every eligible plane drawing, allowing curved edges as well as these straight segments.[1]

The general plane case remains open. Some special kinds of drawings and weaker edge bounds are known. The cited paper finds counterexamples to an extension on surfaces with handles; that changes the surface and does not give a counterexample in the plane.[1]

The five-point star demonstrates equality, not an upper bound for all thrackles. The cited preprint distinguishes the still-open plane inequality from its false extension to surfaces of positive genus. Its constructions there do not change the original drawing rules or resolve the plane case.[1]

Crossings usually offer room to draw a dense network, but requiring every pair to meet exactly once is a much tighter rule. It imposes consistency across all the connections, not just at each dot. The problem studies how a local rule about pairs of curves can constrain the total size of a network.[1]

A thrackle permits many crossings, yet asks for a global arrangement of curves whose pairwise meetings are all compatible. This links combinatorial sparsity to the topology of the plane. The star shows that crossings themselves do not force an edge count below the vertex count, while the other-surface constructions show that the surrounding space affects how densely this same meeting rule can be realized.[1]