The Kelmans-Seymour conjecture for apex graphs

Details The Kelmans-Seymour conjecture for apex graphs The Kelmans-Seymour conjecture for apex graphs

2010-12-28

arxiv.org/abs/1012.5793v1

Mathematics

Combinatorics

10 pages, submitted on August 10th 2010

Scientific paper

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.

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