On the Kuratowski graph planarity criterion

TL;DR

This paper presents a simplified proof of the Kuratowski graph planarity criterion, making it more accessible for students and mathematicians by providing clear definitions and related results.

Contribution

It offers the possibly simplest proof of the Kuratowski graph planarity criterion, with detailed explanations and simplifications from previous proofs.

Findings

Simplified proof of the Kuratowski criterion

Accessible presentation for students and mathematicians

Includes related results on graphs and spaces

Abstract

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.