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Subdivisions in apex graphs

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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.

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Authors and Affiliations

  1. Department of Mathematics, Hamburg University, 20146, Hamburg, Germany

    Elad Aigner-Horev

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  1. Elad Aigner-Horev
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Correspondence to Elad Aigner-Horev.

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Communicated by R. Diestel.

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Aigner-Horev, E. Subdivisions in apex graphs. Abh. Math. Semin. Univ. Hambg. 82, 83–113 (2012). https://doi.org/10.1007/s12188-012-0063-x

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  • DOI: https://doi.org/10.1007/s12188-012-0063-x

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