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