{
  "id": "2404.12204",
  "version": "v1",
  "published": "2024-04-18T14:09:35.000Z",
  "updated": "2024-04-18T14:09:35.000Z",
  "title": "Minimum saturated graphs for unions of cliques",
  "authors": [
    "Wen-Han Zhu",
    "Rong-Xia Hao",
    "Zhen He"
  ],
  "comment": "7 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $H$ be a fixed graph. A graph $G$ is called {\\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\\it saturation number} of $H$, denoted $sat(n,H)$, is the minimum number of edges over all $H$-saturated graphs of order $n$, and $Sat(n,H)$ denote the family of $H$-saturated graphs with $sat(n,H)$ edges and $n$ vertices. In this paper, we resolve a conjecture of Chen and Yuan in[Discrete Math. 347(2024)113868] by determining $Sat(n,K_p\\cup (t-1)K_q)$ for every $2\\le p\\le q$ and $t\\ge 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-18T14:09:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "minimum saturated graphs",
      "minimum number",
      "discrete math",
      "saturation number",
      "missing edge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}