The Kelmans-Seymour conjecture IV: a proof

TL;DR

This paper proves the Kelmans-Seymour conjecture, establishing that graphs without a subdivision of $K_5$ are either planar or have a small cut, advancing understanding of graph structure and planarity conditions.

Contribution

The paper provides a complete proof of the long-standing Kelmans-Seymour conjecture, confirming the structural property of graphs lacking a $K_5$ subdivision.

Findings

Proof of the Kelmans-Seymour conjecture.

Graphs without $K_5$ subdivision are planar or have a cut of size at most 4.

Discussion of related results and open problems.

Abstract

A well known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of $K_5$ or $K_{3,3}$. Wagner proved in 1937 that if a graph other than $K_5$ does not contain any subdivision of $K_{3,3}$ then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of $K_5$ then it is planar or it admits a cut of size at most 4. In this paper, we give a proof of the Kelmans-Seymour conjecture. We also discuss several related results and problems.