Paper detail

On the Kuratowski graph planarity criterion

This paper is purely expositional. The statement of the Kuratowski graph planarity criterion is simple and well-known. However, its classical proof is not easy. In this paper we present the Makarychev proof (with further simplifications by Prasolov, Telishev, Zaslavski and the author) which is possibly the simplest. In the Rusian version before the proof we present all the necessary definitions, and afterwards we state some close results on graphs and more general spaces. The paper is accessible for students familiar with the notion of a graph, and could be an interesting easy reading for mature mathematicians.