The Kelmans-Seymour conjecture for apex 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.

10 pages, submitted on August 10th 2010