{
  "id": "2107.00916",
  "version": "v1",
  "published": "2021-07-02T09:11:31.000Z",
  "updated": "2021-07-02T09:11:31.000Z",
  "title": "The fractional chromatic number of $K_Δ$-free graphs",
  "authors": [
    "Xiaolan Hu",
    "Xing Peng"
  ],
  "comment": "19 pages, Comments are welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a simple graph $G$, let $\\chi_f(G)$ be the fractional chromatic number of $G$. In this paper, we aim to establish upper bounds on $\\chi_f(G)$ for those graphs $G$ with restrictions on the clique number. Namely, we prove that for $\\Delta \\geq 4$, if $G$ has maximum degree at most $\\Delta$ and is $K_{\\Delta}$-free, then $\\chi_f(G) \\leq \\Delta-\\tfrac{1}{8}$ unless $G= C^2_8$ or $G = C_5\\boxtimes K_2$. This im proves the result in [King, Lu, and Peng, SIAM J. Discrete Math., 26(2) (2012), pp. 452-471] for $\\Delta \\geq 4$ and the result in [Katherine and King, SIAM J.Discrete Math., 27(2) (2013), pp. 1184-1208] for $\\Delta \\in \\{6,7,8\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-02T09:11:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C72",
      "05C15"
    ],
    "keywords": [
      "fractional chromatic number",
      "free graphs",
      "discrete math",
      "simple graph",
      "clique number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}