Let $G = (V, E)$ be a simple graph that can be embedded on a torus so that every region is bordered by exactly $3$ edges. Find (with a proof) all the possible values for the quantity $|V | − |E| + r$, where $r$ is the number of regions that the graph $G$ can split the torus.
So the torus will be covered in $C_3$'s connected to each other. I'm not really sure what to do with that information. Any help/hints will be appreciated.