{
  "id": "2308.15248",
  "version": "v1",
  "published": "2023-08-29T12:14:06.000Z",
  "updated": "2023-08-29T12:14:06.000Z",
  "title": "On the chromatic number of some ($P_3\\cup P_2$)-free graphs",
  "authors": [
    "Rui Li",
    "Jinfeng Li",
    "Di Wu"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:2308.05442, arXiv:2308.08768",
  "categories": [
    "math.CO"
  ],
  "abstract": "A hereditary class $\\cal G$ of graphs is {\\em $\\chi$-bounded} if there is a {\\em $\\chi$-binding function}, say $f$, such that $\\chi(G)\\le f(\\omega(G))$ for every $G\\in\\cal G$, where $\\chi(G)(\\omega(G))$ denotes the chromatic (clique) number of $G$. It is known that for every $(P_3\\cup P_2)$-free graph $G$, $\\chi(G)\\le \\frac{1}{6}\\omega(G)(\\omega(G)+1)(\\omega(G)+2)$ \\cite{BA18}, and the class of $(2K_2, 3K_1)$-free graphs does not admit a linear $\\chi$-binding function\\cite{BBS19}. In this paper, we prove that (\\romannumeral 1) $\\chi(G)\\le2\\omega(G)$ if $G$ is ($P_3\\cup P_2$, kite)-free, (\\romannumeral 2) $\\chi(G)\\le\\omega^2(G)$ if $G$ is ($P_3\\cup P_2$, hammer)-free, (\\romannumeral 3) $\\chi(G)\\le\\frac{3\\omega^2(G)+\\omega(G)}{2}$ if $G$ is ($P_3\\cup P_2, C_5$)-free. Furthermore, we also discuss $\\chi$-binding functions for $(P_3\\cup P_2, K_4)$-free graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-29T12:14:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free graph",
      "chromatic number",
      "binding function",
      "hereditary class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}