{
  "id": "2408.11644",
  "version": "v1",
  "published": "2024-08-21T14:16:17.000Z",
  "updated": "2024-08-21T14:16:17.000Z",
  "title": "The saturation number for unions of four cliques",
  "authors": [
    "Ruo-Xuan Li",
    "Rong-Xia Hao",
    "Zhen He",
    "Wen-Han Zhu"
  ],
  "comment": "17pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ but $H$ is a subgraph of $G + e$ for any edge $e$ in $\\overline{G}$. The saturation number $sat(n,H)$ for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. The $sat(n, K_{p_1} \\cup K_{p_2} \\cup K_{p_3})$ with $p_3 \\ge p_1 + p_2$ was given in [Discrete Math. 347 (2024) 113868]. In this paper, $sat(n,K_{p_1} \\cup K_{p_2} \\cup K_{p_3} \\cup K_{p_4})$ with $p_{i+1} - p_i \\ge p_1$ for $2 \\le i\\le 3$ and $4\\le p_1\\le p_2$ is determined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-21T14:16:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "saturation number",
      "minimal number",
      "discrete math",
      "saturated graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}