{
  "id": "1907.03913",
  "version": "v1",
  "published": "2019-07-09T00:09:55.000Z",
  "updated": "2019-07-09T00:09:55.000Z",
  "title": "Independent Sets in n-vertex k-chromatic, \\ell-connected graphs",
  "authors": [
    "John Engbers",
    "Lauren Keough",
    "Taylor Short"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally address maximizing the number of independent sets of size 2 (equivalently, minimizing the number of edges) over all $n$-vertex $k$-chromatic $\\ell$-connected graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-09T00:09:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C69"
    ],
    "keywords": [
      "independent sets",
      "n-vertex k-chromatic",
      "connected graphs",
      "unique extremal graph",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}