For example there is the Circular Embedding Conjecture. Every 2-connected graph has an embedding in some surface such that each face is bounded by a simple cycle. The face boundaries form a cycle double cover. The two conjectures are equivalent for cubic graphs, but the second is stronger for noncubic graphs. Likewise one could require the embedding to be in an orientable surface.
Partial results are too numerous to mention here.
Vol 27 Annals of Discrete Math.