Embeddings into the plane of graphs with vertices of degree 4

TL;DR

This paper provides an accessible proof of Vassiliev's conjecture on the planarity of degree-4 graphs, making the result understandable to high-school students and clarifying previous complex proofs.

Contribution

It offers a simplified, clear exposition of the proof of Vassiliev's conjecture, enhancing accessibility for beginners and educational purposes.

Findings

Proof of Vassiliev's conjecture on graph planarity

Accessible explanation suitable for high-school students

Clarification and comments on Manturov's original proof

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.