I'm trying to prove that given a planar graph (by that I mean a graph where every pair of points is joined without crossings) $V-E+F = 2$.
I can prove this by induction directly on the edges except I'm trying to understand how it is true via the Euler characteristic for the sphere.
I can see that a graph can be drawn on the plane in this way if and only if it can be drawn on the sphere in this way (i.e no crossings). Just because the plane is topologically the sphere with a point removed.
Now I'm wondering if it is possible that in some way the surface formed by the graph on the sphere is homeomorphic to the sphere? In which case we would have $\chi = 2$ immediately.
Thanks
When you draw the graph on the sphere, you don't change any topological data, you just decide which points are vertices and which lie on edges. (If you want to see this as a cellular decomposition, you could think of the inclusion of each vertex, edge and face into the sphere, or something similar depending on your precise definition of cellular decomposition, as the maps explaining how your cell complex is built.) This means there is no possibility of the result not being a sphere, and so Euler's formula applies directly.