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.



Koebe–Andreev–Thurston theorem - where can I find a proof?
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.


