Embeddings into the plane of graphs with vertices of degree 4

Abstract

In this expository note we present a proof of the V.A. Vassiliev conjecture on the planarity of graphs with vertices of degree 4 and certain additional structure. Both statement and proof are accessible to high-school students familiar with basic notions of graph theory. The conjecture was first proved by V.O. Manturov (such a proof was one of the main results of his habilitation thesis). In this note the exposition is made clearer and some comments for beginners are added.