Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two discs are tangent iff the corresponding vertices are connected to each other.
Where can I find the/a proof of this theorem, and what should I learn to understand it?
I prefer proofs which are elementary, but other proofs are welcome too.
Best Answer
There are many proofs, and I'm not claiming that the following list is complete. New references are welcome.
(First proof)
(Thurston's rediscovery and related)
(Variational principle)
(An inductive proof ?)
(I also recommend the following completion of the theorem)