The Euler number of any graph embedded in a surface is V+F-E, where V is the number of vertices, F is the number of faces or regions and E is the number of edges.  Note that to compute an Euler number, each region must either be homeomorphic to a disk or, in the case of the plane, be homeomorphic to a punctured disk.

Proved in class: The Euler number of any planar graph is 2.

1. Find the Euler number of a graph embedded in a torus.

2. Find the Euler number of a graph embedded in a 2-holed torus.

3. Prove by induction that the Euler number of every graph embedded in a torus  is the same as  the number you found in problem 1.