---
tag: 30V6WXQZ5F
kind: statement
role: proposition
name: Circular embedding conjecture
sources:
  - wikipedia: en:21486302@r1365384326
    locator: Cycle double cover, section Circular embedding conjecture
    note: adopted
    mention: wikipedia://en/21486302#V7RP
    sections: [Statement]
inventory: [MH3TFY5CN9]
---

## Statement

If a graph has a cycle double cover, the cycles of the cover can be used to form the 2-cells of a graph embedding onto a two-dimensional cell complex. In the case of a cubic graph, this complex always forms a manifold. The graph is said to be *circularly embedded* onto the manifold, in that every face of the embedding is a simple cycle in the graph. However, a cycle double cover of a graph with degree greater than three may not correspond to an embedding on a manifold: the cell complex formed by the cycles of the cover may have non-manifold topology at its vertices. The **circular embedding conjecture** or **strong embedding conjecture** states that every 2-vertex-connected graph has a circular embedding onto a manifold. If so, the graph also has a cycle double cover, formed by the faces of the embedding.

For cubic graphs, 2-vertex-connectedness and bridgelessness are equivalent. Therefore, the circular embedding conjecture is clearly at least as strong as the cycle double cover conjecture.

If a circular embedding exists, it might not be on a surface of minimal genus: Nguyen Huy Xuong described a 2-vertex-connected toroidal graph none of whose circular embeddings lie on a torus.