{
  "id": "1501.00370",
  "version": "v1",
  "published": "2015-01-02T09:22:39.000Z",
  "updated": "2015-01-02T09:22:39.000Z",
  "title": "The coloring of the regular graph of ideals",
  "authors": [
    "Farzad Shaveisi"
  ],
  "comment": "AMS-LaTeX, 11 pages with no figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The regular graph of ideals of the commutative ring $R$, denoted by $\\Gamma_{reg}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is shown that for every Artinian ring $R$, the edge chromatic number of $\\Gamma_{reg}(R)$ equals its maximum degree. Then a formula for the clique number of $\\Gamma_{reg}(R)$ is given. Also, it is proved that for every reduced ring $R$ with $n(\\geq3)$ minimal prime ideals, the edge chromatic number of $\\Gamma_{reg}(R)$ is $2^{n-1}-2$. Moreover, we show that both of the clique number and vertex chromatic number of $\\Gamma_{reg}(R)$ are $n-1$, for every reduced ring $R$ with $n$ minimal prime ideals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-02T09:22:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C25",
      "13B30",
      "16P20",
      "G.2.2"
    ],
    "keywords": [
      "regular graph",
      "minimal prime ideals",
      "edge chromatic number",
      "regular element",
      "clique number"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150100370S"
    }
  }
}