{
  "id": "1209.3899",
  "version": "v4",
  "published": "2012-09-18T10:10:07.000Z",
  "updated": "2013-01-04T08:48:59.000Z",
  "title": "Two sufficient conditions for the existence of Hamilton cycles in graphs",
  "authors": [
    "Bo Ning",
    "Bing Chen",
    "Shenggui Zhang"
  ],
  "comment": "Withdraw this preprint since the results will be added to arxiv:1212.6466",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph on $n\\geq 3$ vertices, claw the bipartite graph $K_{1,3}$, and $Z_i$ the graph obtained from a triangle by attaching a path of length $i$ to its one vertex. $G$ is called 1-heavy if at least one end vertex of each induced claw of $G$ has degree at least $n/2$, and claw-\\emph{o}-heavy if each induced claw of it has a pair of end vertices with degree sum at least $n$. In this paper we prove two results: (1) Every 2-connected claw-$o$-heavy graph $G$ is Hamiltonian if every pair of vertices $u,v$ in a subgraph $H\\cong Z_1$ contained in an induced subgraph $Z_2$ of $G$ with $d_{H}(u,v)=2$ satisfies one of the following conditions: ($a$) $|N(u)\\cap N(v)|\\geq 2$; ($b$) $\\max(d(u),d(v))\\geq n/2$. (2) Every 3-connected 1-heavy graph $G$ is Hamiltonian if every pair of vertices $u,v$ in an induced subgraph $H\\cong Z_2$ of $G$ with $d_{H}(u,v)=2$ satisfies one of the following conditions: ($a$) $|N(u)\\cap N(v)|\\geq 2$; ($b$) $\\max(d(u),d(v))\\geq n/2$. Our results improve or extend previous theorems of Broersma et al., Chen et al., Fan, Goodman & Hedetniemi, Gould & Jacobson and Shi on the existence of Hamilton cycles in graphs.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-01-04T08:48:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamilton cycles",
      "sufficient conditions",
      "induced subgraph",
      "induced claw",
      "degree sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.3899N"
    }
  }
}