Rather than grouping charges in faces of the graph, we can give a dual argument that groups charges at vertices. This proof works best with the convex planar embedding of the graph of a polyhedron, its Schlegel diagram. The proof by projected Schlegel diagrams is closely related, but rearranges charges differently.
Rotate the graph if necessary so that no edge is vertical. As in the previous proof, put a unit \(+\) charge at each vertex, a unit \(-\) charge at the center of each edge, and a unit \(+\) charge in the middle of each face. We will show that all but two \(+\) charges cancel. To do this, displace the charge on each edge to its right endpoint; displace the charge on each face (except the outer face) to its rightmost vertex. Each vertex (except the leftmost vertex) receives the charges from an alternating sequence of edges and faces, cancelling its initial charge. The only remaining uncancelled charges are one \(+\) charge on the outer face and one \(+\) charge on the leftmost vertex.
Proofs of Euler's Formula.
From the Geometry Junkyard,
computational
and recreational geometry pointers.
David Eppstein,
Theory Group,
ICS,
UC Irvine.