{
  "id": "1103.2685",
  "version": "v6",
  "published": "2011-03-14T15:07:18.000Z",
  "updated": "2014-10-27T13:40:07.000Z",
  "title": "Ramsey numbers for trees",
  "authors": [
    "Zhi-Hong Sun"
  ],
  "comment": "10 pages",
  "journal": "Bull. Aust. Math. Soc. 86(2012), no.1, 164-176",
  "categories": [
    "math.CO"
  ],
  "abstract": "For $n\\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $\\Delta(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\\{v_0,v_1,\\ldots,$ $v_{n-1}\\}$ and $E=\\{v_0v_1,\\ldots,v_0v_{n-3},$ $v_{n-3}v_{n-2},v_{n-2}v_{n-1}\\}$. In this paper we evaluate the Ramsey numbers $r(G_m,T_n')$ and $r(G_m,T_n^*)$, where $G_m$ is a connected graph of order $m$. As examples, for $n\\ge 8$ we have $r(T_n',T_n^*)=r(T_n^*,T_n^*)=2n-5$, for $n>m\\ge 7$ we have $r(K_{1,m-1},T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\\mid (n-3)$ or $m-1\\nmid (n-3)$, for $m\\ge 7$ and $n\\ge (m-3)^2+2$ we have $r(T_m^*,T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\\mid (n-3)$ or $m-1\\nmid (n-3)$.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2011-10-25T13:02:31.000Z",
      "abstract": "For $n\\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $\\Delta(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\\{v_0,v_1,...,$ $v_{n-1}\\}$ and $E=\\{v_0v_1,...,v_0v_{n-3},$ $v_{n-3}v_{n-2},v_{n-2}v_{n-1}\\}$. In this paper we evaluate the Ramsey numbers $r(G_m,T_n')$ and $r(G_m,T_n^*)$, where $G_m$ is a connected graph of order $m$. As examples, for $n\\ge 8$ we have $r(T_n',T_n^*)=r(T_n^*,T_n^*)=2n-5$, for $n>m\\ge 7$ we have $r(K_{1,m-1},T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\\mid (n-3)$ or $m-1\\nmid (n-3)$, for $m\\ge 7$ and $n\\ge (m-3)^2+2$ we have $r(T_m^*,T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\\mid (n-3)$ or $m-1\\nmid (n-3)$.",
      "comment": "11 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v6",
      "updated": "2014-10-27T13:40:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C05",
      "05C55"
    ],
    "keywords": [
      "ramsey numbers",
      "unique tree",
      "connected graph"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.2685S"
    }
  }
}