Circular embedding conjecture

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.