The Kelmans-Seymour conjecture for apex graphs

2010/12/28 by Aigner-Horev, Elad, Krakovski, Roi
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1012.5793

Abstract

We provide a short proof that a 5-connected nonplanar apex graph contains a subdivided K₅ or a K^-₄ (= K₄ 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 K₅; this settles the Kelmans-Seymour conjecture for apex graphs.

Related

• Subdivisions in apex graphs
• The Kelmans-Seymour conjecture I: special separations
• The Kelmans-Seymour conjecture IV: a proof
• The Kelmans-Seymour conjecture III: 3-vertices in K₄^-
• The Kelmans-Seymour conjecture II: 2-vertices in K₄^-