{
  "id": "1101.5278",
  "version": "v1",
  "published": "2011-01-27T13:12:28.000Z",
  "updated": "2011-01-27T13:12:28.000Z",
  "title": "Nonseparating K4-subdivisions in graphs of minimum degree at least 4",
  "authors": [
    "Matthias Kriesell"
  ],
  "comment": "IMADA-preprint-math, 25 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We first prove that for every vertex x of a 4-connected graph G there exists a subgraph H in G isomorphic to a subdivision of the complete graph K4 on four vertices such that G-V(H) is connected and contains x. This implies an affirmative answer to a question of W. Kuehnel whether every 4-connected graph G contains a subdivision H of K4 as a subgraph such that G-V(H) is connected. The motor for our induction is a result of Fontet and Martinov stating that every 4-connected graph can be reduced to a smaller one by contracting a single edge, unless the graph is the square of a cycle or the line graph of a cubic graph. It turns out that this is the only ingredience of the proof where 4-connectedness is used. We then generalize our result to connected graphs of minimum degree at least 4, by developing the respective motor: A structure theorem for the class of simple connected graphs of minimum degree at least 4.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-01-27T13:12:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimum degree",
      "nonseparating k4-subdivisions",
      "complete graph k4",
      "cubic graph",
      "line graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.5278K"
    }
  }
}