The Kelmans-Seymour conjecture for apex graphs

TL;DR

This paper proves that 5-connected nonplanar apex graphs contain specific subgraphs, confirming the Kelmans-Seymour conjecture for this class of graphs through a concise proof and leveraging recent related results.

Contribution

It provides a short proof confirming the Kelmans-Seymour conjecture for apex graphs, combining new proof techniques with recent findings.

Findings

A 5-connected nonplanar apex graph contains a subdivided K5 or a K4 minus an edge.

The proof simplifies previous approaches to the conjecture for apex graphs.

The result completes the conjecture's validation for this class of graphs.

Abstract

We provide a short proof that a 5-connected nonplanar apex graph contains a subdivided $K_{_5}$ or a $K^-_{_4}$ (= $K_{_4}$ with a single edge removed) as a subgraph. Together with a recent result of Ma and Yu that {\sl every nonplanar 5-connected graph containing $K^-_{_4}$ as a subgraph has a subdivided $K_{_5}$}; this settles the Kelmans-Seymour conjecture for apex graphs.