Kuratowski's theorem

Abstract

We present three short proofs of Kuratowski's theorem on planarity of graphs and discuss applications, extensions, and some related problems. INTRODUCTIONPlanar graphs are of great importance in graph theory. They are interesting in their own right and their chromatic, enumerative, hamiltonian, and other properties have been studied in great detail. Furthermore, planar graphs are of some importance in the study of convex polytopes since, by Steinitz's theorem, the 1-skeletons of the 3-dimensional polytopes are precisely the 3-connected planar graphs. Finally, planar graphs provide an important link between graphs and matroids.In The importance of Kuratowski's theorem is not so much its applications to planar graph theory (in fact, it is used only in relatively few results on planar graphs). It rather lies in the fact that a characterization of planar graphs in terms of essentially a finite number of forbidden subgraphs exists at all. Also, Kuratowski's theorem has a special ranking among the known planarity criteria not only because of its beauty and simplicity, but also because it implies rather easily the planarity criteria by MacLane [27] ([40]). It should also be noted that, unlike most other planarity criteria, Kuratowski's theorem provides a useful characterization of the nonplanar graphs. Finally, it seems that almost all known proofs of Kuratowski's theorem can be turned into polynomially

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