Koebe–Andreev–Thurston theorem - where can I find a proof? 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.
 A: This is proved in sections 13.6 and 13.7 of Thurston's notes. 
See also Corollary 5.10.4 and section 5.11 of Thurston's book. Another short proof appears in Appendix 2 of Rodin and Sullivan. 
A: EDIT. There are very many published proofs.
For example, Colin de Verdiere (Forum Math., 1 (1989) 395-402), or
Charles R. Collins and Kenneth Stephenson, A circle packing algorithm, Computational Geometry 25 (2003) 233–256.
Thurston's original proof is elementary: he gives an algorithm for finding the
(essentially unique) circle packing from a triangulation. He never published this properly;
his "Geometry and Topology of 3-manifolds" exists as a preprint, and the proof there are sketchy.
Then Colin de Verdiere (Forum Math., 1 (1989) 395-402) proved accurately convergence of Thurston's algorithm.
Another proof of convergence and implementation on computer is in the paper of Collins and Stephenson mentioned above.
Then there were many generalizations and versions of this theorem and algorithm, too many to list all of them here.  
A: There are many proofs, and I'm not claiming that the following list is complete.  New references are welcome.
(First proof)


*

*Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs. Akad. Leipzig 88 (1936), 141–164 (German)


(Thurston's rediscovery and related)


*

*Andreev, E. M., Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (1970), no. 125, 256–260.

*(see also) Roeder, Roland K.W., Hubbard, John H. and Dunbar, William D., Andreev’s Theorem on hyperbolic polyhedra, Annales de l’institut Fourier 57 (2007), no. 3, 825–882. 

*William P. Thurston and John W. Milnor, The Geometry and Topology of Three-Manifolds


(Variational principle)


*

*Yves Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), no. 3, 655–669 (French).

*Igor Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994), 553–580.

*Alexander I. Bobenko and Boris A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689.

*(see also) Günter M. Ziegler, Convex polytopes: extremal constructions and f-vector shapes, Geometric Combinatorics, 2007, pp. 617–691.


(An inductive proof ?)


*

*Kenneth Stephenson, Introduction to Circle Packing: The theory of discrete analytic functions,
Cambridge University Press, Cambridge, 2005.


(I also recommend the following completion of the theorem)


*

*Graham R. Brightwell and Edward R. Scheinerman, Representations of planar graphs, SIAM J. Discrete Math. 6 (1993), no. 2, 214–229.

