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  "body": "\n## Statement\n\nIf 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.\n\nFor 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.\n\nIf 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.\n",
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