The Kelmans-Seymour conjecture for apex graphs
arXiv:1012.5793
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.
10 pages, submitted on August 10th 2010