{
  "id": "2205.14556",
  "version": "v1",
  "published": "2022-05-29T02:24:45.000Z",
  "updated": "2022-05-29T02:24:45.000Z",
  "title": "Tight bounds on a conjecture of Gallai",
  "authors": [
    "Jun Gao",
    "Jie Ma"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that for $n>k\\geq 3$, if $G$ is an $n$-vertex graph with chromatic number $k$ but any its proper subgraph has smaller chromatic number, then $G$ contains at most $n-k+3$ copies of cliques of size $k-1$. This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-29T02:24:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tight bound",
      "conjecture",
      "smaller chromatic number",
      "proper subgraph",
      "vertex graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}