Euler's Formula, Proof 4: Induction on Edges

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.