{
  "id": "1504.06585",
  "version": "v1",
  "published": "2015-04-24T18:04:45.000Z",
  "updated": "2015-04-24T18:04:45.000Z",
  "title": "Clique number of the square of a line graph",
  "authors": [
    "Małgorzata Śleszyńska-Nowak"
  ],
  "comment": "8 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "An edge coloring of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $\\chi _{s}^{\\prime }(G)$, is the minimum number of colors for which $G$ has a strong edge coloring. The strong chromatic index of $G$ is equal to the chromatic number of the square of the line graph of $G$. The chromatic number of the square of the line graph of $G$ is greater than or equal to the clique number of the square of the line graph of $G$, denoted by $\\omega(L)$. In this note we prove that $\\omega(L) \\le 1.5 \\Delta_{G}^2$ for every graph $G$. Our result allows to calculate an upper bound for the fractional strong chromatic index of $G$, denoted by $\\chi_{fs}^\\prime(G)$. We prove that $\\chi_{fs}^{\\prime}(G) \\le 1.75 \\Delta_G^2$ for every graph $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-24T18:04:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "line graph",
      "clique number",
      "chromatic number",
      "fractional strong chromatic index",
      "edge coloring"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}