The Kelmans-Seymour conjecture I: special separations

2015/11/16 by He, Dawei, Wang, Yan, Yu, Xingxing
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1511.05020

Abstract

Seymour and, independently, Kelmans conjectured in the 1970s that every 5-connected nonplanar graph contains a subdivision of K₅. This conjecture was proved by Ma and Yu for graphs containing K₄^-, and an important step in their proof is to deal with a 5-separation in the graph with a planar side. In order to establish the Kelmans-Seymour conjecture for all graphs, we need to consider 5-separations and 6-separations with less restrictive structures. The goal of this paper is to deal with special 5-separations and 6-separations, including those with an apex side. Results will be used in subsequent papers to prove the Kelmans-Seymour conjecture.

Related

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