{
  "id": "2210.04682",
  "version": "v2",
  "published": "2022-10-10T13:32:17.000Z",
  "updated": "2023-04-10T08:41:52.000Z",
  "title": "The chromatic number of ($P_{5}, K_{5}-e$)-free graphs",
  "authors": [
    "Yian Xu"
  ],
  "comment": "This paper needs to be rewrote and reorganized. The last section might not be fully correct, it needs some further check",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph. We use $\\chi(G)$ and $\\omega(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices. A family of graphs $\\mathcal{G}$ is said to be {\\it$\\chi$-bounded} if there exists some function $f$ such that $\\chi(G)\\leq f(\\omega(G))$ for every $G\\in\\mathcal{G}$. In this paper, we show that the family of $(P_5, K_5-e)$-free graphs is $\\chi$-bounded by a linear function: $\\chi(G)\\leq \\max\\{13,\\omega(G)+1\\}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-04-10T08:41:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free graphs",
      "chromatic number",
      "linear function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}