## Abstract

The Kelmans-Seymour conjecture states that every 5-connected nonplanar graph contains a subdivided K _{5}. Certain questions of Mader propose a "plan" towards a possible resolution of this conjecture. One part of this plan is to show that every 5-connected nonplanar graph containing K _{4} ^{-} or K _{2,3} as a subgraph has a subdivided K _{5}. Recently, Ma and Yu showed that every 5-connected nonplanar graph containing K _{4} ^{-} as a subgraph has a subdivided K _{5}. We take interest in K _{2,3} and prove that every 5-connected nonplanar apex graph containing K _{2,3} as a subgraph contains a subdivided K _{5}. The result of Ma and Yu can be used in a short discharging argument to prove that every 5-connected nonplanar apex graph contains a subdivided K _{5}; here we propose a longer proof whose merit is that it avoids the use of discharging and employs a more structural approach; consequently it is more amenable to generalization.

Original language | English |
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Pages (from-to) | 83-113 |

Number of pages | 31 |

Journal | Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg |

Volume | 82 |

Issue number | 1 |

DOIs | |

State | Published - Apr 2012 |

Externally published | Yes |

## Keywords

- (Rooted) subdivisions
- Apex graphs
- Nonseparating paths