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 log^4 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.