A cycle double cover of an undirected graph is a collection of cycles that cover each edge of the graph exactly twice.
For a polyhedral graph, the faces of a corresponding convex polyhedron give a double cover of the graph since each edge belongs to exactly two faces. As an example, the cycle double cover of the cubical graph is illustrated above.
The cycle double cover conjecture states that every bridgeless graph has a cycle double cover. In July 2026, OpenAI released a claimed AI-generated proof of the conjecture (OpenAI 2026).
