Hello all,

First of all I hope I am in the right place for a Graph Theory + Topology question.

My problem is as follows:

I have found (in my research - I haven't proven it yet) that the Euler Characteristic of the 1-Torus is 0. So V-E+F=0. But I can't get that to work with examples. For example, one placement of k4 on a torus has V=4 and E=4, and therefore supposedly has F=2, but that would mean that the face with a hole in it counts as -1 face! Also, if I have one point on the other side of the torus, counting regions becomes very confusing (and I'm still not able to get the result of 0 as desired). Even if I draw the torus with its planar representation (the rectangle with corresponding opposite sides) I am still not able to get nice results.

Does anyone have any suggestions how I can proceed?

Any help would be greatly appreciated.