I originally didn't do this right, so our professor told us to try it over and see if we can figure it out. The question is:
a) Can a triangulation (using only triangles) of the 2-holed torus have exactly 6 edges coming in to each vertex? Explain. Hint: Recall that in class we showed that E = 3F/2 (after gluing) for triangulations by triangles.
b) Exactly 7 edges? Explain.
c) How many triangles will be needed in the triangulation in b)?
Another hint he gave us was: Fb (before gluing) = Fa (after gluing) = F. (In other words, the # of faces doesn't change after gluing) The hint is to compare:
F vs. Vb and F vs. Va