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- Publisher
- Springer Journals
- Copyright
- Copyright © 2012 by Mathematisches Seminar der Universität Hamburg and Springer
- Subject
- Mathematics; Algebra; Differential Geometry; Combinatorics; Geometry; Number Theory; Topology
- ISSN
- 0025-5858
- eISSN
- 1865-8784
- DOI
- 10.1007/s12188-012-0063-x
- Publisher site
- See Article on Publisher Site
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.
Journal
Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals
Published: Jan 17, 2012