{
  "id": "1212.3898",
  "version": "v1",
  "published": "2012-12-17T06:36:15.000Z",
  "updated": "2012-12-17T06:36:15.000Z",
  "title": "On coloring of fractional powers of graphs",
  "authors": [
    "Stephen Hartke",
    "Hong Liu",
    "Šárka Petříčková"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For $m, n\\in \\N$, the fractional power $\\Gmn$ of a graph $G$ is the $m$th power of the $n$-subdivision of $G$, where the $n$-subdivision is obtained by replacing each edge in $G$ with a path of length $n$. It was conjectured by Iradmusa that if $G$ is a connected graph with $\\Delta(G)\\ge 3$ and $1<m<n$, then $\\chi(\\Gmn)=\\omega(\\Gmn)$. Here we show that the conjecture does not hold in full generality by presenting a graph $H$ for which $\\chi(H^{3/5})>\\omega(H^{3/5})$. However, we prove that the conjecture is true if $m$ is even. We also study the case when $m$ is odd, obtaining a general upper bound $\\chi(\\Gmn)\\leq \\omega(\\Gmn)+2$ for graphs with $\\Delta(G)\\geq 4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-17T06:36:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "fractional power",
      "conjecture",
      "general upper bound",
      "th power",
      "full generality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.3898H"
    }
  }
}