{
  "id": "1503.03272",
  "version": "v1",
  "published": "2015-03-11T11:10:24.000Z",
  "updated": "2015-03-11T11:10:24.000Z",
  "title": "On the existence of vertex-disjoint subgraphs with high degree sums",
  "authors": [
    "Shuya Chiba",
    "Nicolas Lichiardopol"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a graph $G$, we denote by $\\sigma_{2}(G)$ the minimum degree sum of two non-adjacent vertices if $G$ is non-complete; otherwise, $\\sigma_{2}(G) = +\\infty$. In this paper, we prove the following two results; (i) If $s_{1}$ and $s_{2}$ are integers with $s_{1}, s_{2} \\ge 2$ and if $G$ is a non-complete graph with $\\sigma_{2}(G) \\ge 2(s_{1} + s_{2} + 1) - 1$, then $G$ contains two vertex-disjoint subgraphs $H_{1}$ and $H_{2}$ such that each $H_{i}$ is a graph of order at least $s_{i}+1$ with $\\sigma_{2}(H_{i}) \\ge 2s_{i} - 1$. (ii) If $s_{1}$ and $s_{2}$ are integers with $s_{1}, s_{2} \\ge 2$ and if $G$ is a non-complete triangle-free graph with $\\sigma_{2}(G) \\ge 2(s_{1} + s_{2}) - 1$, then $G$ contains two vertex-disjoint subgraphs $H_{1}$ and $H_{2}$ such that each $H_{i}$ is a graph of order at least $2s_{i}$ with $\\sigma_{2}(H_{i}) \\ge 2s_{i} - 1$. By using this kind of results, we also give some corollaries concerning the degree conditions for vertex-disjoint cycles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-11T11:10:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "high degree sums",
      "vertex-disjoint subgraphs",
      "non-complete triangle-free graph",
      "minimum degree sum",
      "non-complete graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150303272C"
    }
  }
}