{
  "id": "1902.05381",
  "version": "v1",
  "published": "2019-02-14T14:39:31.000Z",
  "updated": "2019-02-14T14:39:31.000Z",
  "title": "The simple graph threshold number $σ(r,s,a,t)$",
  "authors": [
    "A. J. W. Hilton",
    "A. Rajkumar"
  ],
  "comment": "38 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For $d \\ge 1$, $s \\ge 0$ a $(d, d+s)$-{\\em graph} is a graph whose degrees all lie in the interval $\\{d, d+1, \\ldots, d + s\\}$. For $r \\ge 1$, $a \\ge 0$, an $(r, r+a)$-{\\em factor} of a graph $G$ is a spanning $(r, r+a)$-subgraph of $G$. An $(r, r+a)$-{\\em factorization} of a graph $G$ is a decomposition of $G$ into edge-disjoint $(r, r+a)$-factors. A graph is $(r, r+a)$-{\\em factorable} if it has an $(r, r+a)$-factorization. Let $\\sigma(r, s, a, t)$ be the least integer such that, if $d \\ge \\sigma(r, s, a, t)$, then every $(d, d+s)$-simple graph $G$ is $(r,r+a)$-factorable with $x$ factors for at least $t$ different values of $x$. In this paper we evaluate $\\sigma(r,s,a,t)$ for all values of $r, s, a$ and $t$. We also show that if $a \\ge 2$ and $r \\ge 1$, then, when $r$ is even and $a$ is odd, every $(d, d+s)$-simple graph $G$ has an $(r, r+a)$-factorization with $x$ factors if and only if $$ \\frac{d+s}{r+a}\\, < x \\le \\frac{d}{r}\\,,$$ and we prove similar statements for other parities of $r$ and $a$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-14T14:39:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C70"
    ],
    "keywords": [
      "simple graph threshold number",
      "factorization",
      "similar statements",
      "decomposition",
      "edge-disjoint"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}