By combining the two previous proofs, on induction on faces and induction on vertices we get another induction proof with a much simpler base case.
If the connected planar multigraph \(G\) has no edges, it is an isolated vertex and \(V+F-E=1+1-0=2\). Otherwise, choose any edge \(e\). If \(e\) connects two distinct vertices, contract it, reducing \(V\) and \(E\) by one. Otherwise, \(e\) is a loop connecting one vertex to itself. It is a Jordan curve and separates two faces; remove it and reduce \(F\) and \(E\) by one. In either case the result follows by induction.
This is the proof used by van Lint and Wilson. Other variants are possible – if \(e\) connects two vertices and separates two faces, we may choose in various ways whether to delete or contract \(e\).
Proofs of Euler's Formula.
From the Geometry Junkyard,
computational
and recreational geometry pointers.
David Eppstein,
Theory Group,
ICS,
UC Irvine.