I know that a non-planar graph with one crossing can be embedded in a torus, and I expected that a graph with two crossings would require a double torus. This does not seem to be the case (cf. the Peterson graph), and I am wondering what the relationship is between the crossing number of the graph and the genus of the surface.
Ultimately, I am looking for a way to guide my students to see that the Euler characteristic of a connected graph (when embedded in the simplest suitable surface) is dependent upon the genus of that surface. In other words, that the Euler characteristic tells us more about the surface than about the graph.