For a planar graph $G = (V, E)$ there is the well known bound $|E| \leq 3|V| - 6$. If instead of $S^2$ $G$ embeds in the orientable surface $S_g$ of genus $2 - 2g$ with minimal $g$, what can be said about the number of edges?
If all the steps in the proof I know of for the bound carried over, the bound would only depend on the genus $g$, giving the bound $3|V| - 6 + 6g$ which doesn't seem right.