{
  "id": "2002.11100",
  "version": "v1",
  "published": "2020-02-25T18:55:31.000Z",
  "updated": "2020-02-25T18:55:31.000Z",
  "title": "Clique minors in graphs with a forbidden subgraph",
  "authors": [
    "M. Bucić",
    "J. Fox",
    "B. Sudakov"
  ],
  "comment": "11 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "The classical Hadwiger conjecture dating back to 1940's states that any graph of chromatic number at least $r$ has the clique of order $r$ as a minor. Hadwiger's conjecture is an example of a well studied class of problems asking how large a clique minor one can guarantee in a graph with certain restrictions. One problem of this type asks what is the largest size of a clique minor in a graph on $n$ vertices of independence number $\\alpha(G)$ at most $r$. If true Hadwiger's conjecture would imply the existence of a clique minor of order $n/\\alpha(G)$. Results of Kuhn and Osthus and Krivelevich and Sudakov imply that if one assumes in addition that $G$ is $H$-free for some bipartite graph $H$ then one can find a polynomially larger clique minor. This has recently been extended to triangle free graphs by Dvo\\v{r}\\'ak and Yepremyan, answering a question of Norin. We complete the picture and show that the same is true for arbitrary graph $H$, answering a question of Dvo\\v{r}\\'ak and Yepremyan. In particular, we show that any $K_s$-free graph has a clique minor of order $c_s(n/\\alpha(G))^{1+\\frac{1}{10(s-2) }}$, for some constant $c_s$ depending only on $s$. The exponent in this result is tight up to a constant factor in front of the $\\frac{1}{s-2}$ term.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-25T18:55:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "forbidden subgraph",
      "polynomially larger clique minor",
      "true hadwigers conjecture",
      "triangle free graphs",
      "constant factor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}