Cycle Double Cover conjecture for bridgeless cubic graphs

Prove that every bridgeless cubic graph has a surface embedding with no singular edges, equivalently, an embedding in which no edge lies twice on the boundary of a single face.

Background

The paper identifies the Cycle Double Cover (CDC) conjecture as a central unresolved problem related to embeddings of cubic graphs. In the embedding formulation given by the authors, the conjecture asserts that every bridgeless cubic graph admits a surface embedding without singular edges, where a singular edge is an edge appearing twice on the boundary of one face.

The paper does not resolve the conjecture. Instead, it develops an algorithm for 3-edge-connected cubic graphs that produces a well-spread perfect matching in time O(n log4 n), and uses this result to construct embeddings with at most n/10 singular edges. Thus, the result provides a bounded-singular-edge approximation to the conjectured zero-singular-edge outcome.

References

OpenAI recently announced a proof of the Cycle Double Cover (CDC) Conjecture, which has long stood as one of the central and most challenging open problems in graph theory.

— Some Early Results by Tutte Regarding the Cycle Double Cover Conjecture in 1948  (2609.10618 - Zhang, 8 Sep 2026) in Section 1, “The CDC Conjecture Has Been Open for at Least 80 Years”

In the language of graph embedding, the CDC conjecture is equivalent to every bridgeless cubic graph having a surface embedding with no singular edges (i.e., with no edge that is on the boundary of one face twice).

— On the time complexity of finding a well-spread perfect matching in bridgeless cubic graphs  (2503.00263 - Ghanbari et al., 1 Mar 2025) in Section 1, Introduction