Question: 1. For which m,n is Km,n planar and for which m,n is Km,n non planar? Make and prove a conjecture.
2. Check out Figure for an interesting way to represent the torus (surface of a doughnut):
(a) Convince yourself that the left and right diagrams really are two representations of the same thing.
(b) Draw a few graphs on copies of the torus. (Make sure to cross each of the rectangle edges so that you're genuinely using the torus.)
(c) Try out Euler's formula on these graphs. Does it still hold? If not, does some other formula hold?
(d) Special challenge: Prove your conjecture.
