Subdivisions in apex graphs

摘要

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.