{
  "id": "2408.02400",
  "version": "v1",
  "published": "2024-08-05T11:51:10.000Z",
  "updated": "2024-08-05T11:51:10.000Z",
  "title": "Disproof of a conjecture by Erdős, Gimbel and Straight",
  "authors": [
    "Raphael Steiner"
  ],
  "comment": "4 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The cochromatic number $\\zeta(G)$ of a graph $G$ is the smallest number of colors in a vertex-coloring of $G$ such that every color class induces an independent set or a clique. In 1988, Erd\\H{o}s, Gimbel and Straight conjectured that every $K_5$-free graph $G$ with $\\zeta(G)>3$ satisfies $\\chi(G)\\le \\zeta(G)+2$. In this note, we present a counterexample to this conjecture and discuss related results and questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-05T11:51:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C69"
    ],
    "keywords": [
      "conjecture",
      "color class induces",
      "free graph",
      "cochromatic number",
      "independent set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}