{
  "id": "0907.3705",
  "version": "v3",
  "published": "2009-07-21T19:09:41.000Z",
  "updated": "2010-03-12T23:01:02.000Z",
  "title": "On hitting all maximum cliques with an independent set",
  "authors": [
    "Landon Rabern"
  ],
  "comment": "Added simple proof of Kostochka's lemma.",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that every graph $G$ for which $\\omega(G) \\geq 3/4(\\Delta(G) + 1)$, has an independent set $I$ such that $\\omega(G - I) < \\omega(G)$. It follows that a minimum counterexample $G$ to Reed's conjecture satisfies $\\omega(G) < 3/4(\\Delta(G) + 1)$ and hence also $\\chi(G) > \\lceil 7/6\\omega(G) \\rceil$. We also prove that if for every induced subgraph $H$ of $G$ we have $\\chi(H) \\leq \\max{\\lceil 7/6\\omega(H) \\rceil, \\lceil \\frac{\\omega(H) + \\Delta(H) + 1}{2}\\rceil}$, then we also have $\\chi(G) \\leq \\lceil \\frac{\\omega(G) + \\Delta(G) + 1}{2}\\rceil$. This gives a generic proof of the upper bound for line graphs of multigraphs proved by King et al.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-03-12T23:01:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "independent set",
      "maximum cliques",
      "reeds conjecture satisfies",
      "minimum counterexample",
      "generic proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.3705R"
    }
  }
}