The cycle double cover conjecture for bridgeless graphs

Let $\Gamma$ be a graph. A cycle double cover is a collection of cycles such that every edge of $\Gamma$ lies in exactly two cycles. For a cubic graph, this is equivalent to requiring that exactly three cycles pass through each node. A simplicial surface has $\Gamma$ as its face graph when its faces correspond to the vertices and edges of $\Gamma$ in the usual incidence structure.

Cycle double cover conjecture. Every cubic graph containing no bridges has a cycle double cover, or equivalently is the face graph of a simplicial surface. More generally, every bridgeless graph admits a cycle double cover.

The conjecture would characterize which bridgeless cubic graphs arise as face graphs of simplicial surfaces and remains open: it is not known whether all cubic bridgeless graphs admit a cycle double cover.