A proof that: for any given polyhedron so shaped that every closed non-self intersecting broken line composed of edges of the polyhedron divides the surface of the polyhedron into precisely two disjoint regions each of which is bounded by the closed broken line, v - e + f = 2, where v is the number of vertices of the polyhedron, e the number of edges and f the number of faces.
Harbison, Charles H.
"Faces, Edges, Vertices of Some Polyhedra,"
Journal of the Arkansas Academy of Science: Vol. 28
, Article 10.
Available at: http://scholarworks.uark.edu/jaas/vol28/iss1/10