{
  "id": "1208.4734",
  "version": "v1",
  "published": "2012-08-23T12:10:08.000Z",
  "updated": "2012-08-23T12:10:08.000Z",
  "title": "New approach to the $k$-independence number of a graph",
  "authors": [
    "Yair Caro",
    "Adriana Hansberg"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G = (V,E)$ be a graph and $k \\ge 0$ an integer. A $k$-independent set $S \\subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\\alpha_k(G)$ we denote the maximum cardinality of a $k$-independent set of $G$. We prove that, for a graph $G$ on $n$ vertices and average degree $d$, $\\alpha_k(G) \\ge \\frac{k+1}{\\lceil d \\rceil + k + 1} n$, improving the hitherto best general lower bound due to Caro and Tuza [Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99-107].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-23T12:10:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "independence number",
      "hitherto best general lower bound",
      "independent set",
      "maximum degree",
      "maximum cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.4734C"
    }
  }
}