{
  "id": "1901.06985",
  "version": "v1",
  "published": "2019-01-21T16:18:17.000Z",
  "updated": "2019-01-21T16:18:17.000Z",
  "title": "A note on Hadwiger's Conjecture for $W_5$-free graphs with independence number two",
  "authors": [
    "Christian Bosse"
  ],
  "comment": "7 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Hadwiger number of a graph $G$, denoted $h(G)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h(G) \\ge \\chi(G)$, where $\\chi(G)$ denotes the chromatic number of $G$. Let $\\alpha(G)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h(G) \\ge \\chi(G)$ for all $H$-free graphs $G$ with $\\alpha(G) \\le 2$, where $H$ is any graph on four vertices with $\\alpha(H) \\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell considered a particular strengthening of Hadwiger's conjecture due to Seymour and subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\\alpha(H) \\le 2$. In this note, we prove that $h(G) \\ge \\chi(G)$ for all $W_5$-free graphs $G$ with $\\alpha(G) \\le 2$, where $W_5$ denotes the wheel on six vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-21T16:18:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C83",
      "05C15"
    ],
    "keywords": [
      "free graphs",
      "hadwigers conjecture",
      "independence number",
      "largest integer",
      "hadwiger number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}