{
  "id": "1305.6195",
  "version": "v2",
  "published": "2013-05-27T12:32:29.000Z",
  "updated": "2013-10-03T18:02:36.000Z",
  "title": "Maximum 4-degenerate subgraph of a planar graph",
  "authors": [
    "Robert Lukoťka",
    "Ján Mazák",
    "Xuding Zhu"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \\ge 2$ has a 4-degenerate induced subgraph containing at least $(38-d)/36$ of its vertices. This shows that every planar graph of order $n$ has a 4-degenerate induced subgraph of order more than $8/9 \\cdot n$. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-03T18:02:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "induced subgraph",
      "subsequent removals",
      "connected planar graph",
      "average degree",
      "empty graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.6195L"
    }
  }
}