{
  "id": "2101.00507",
  "version": "v1",
  "published": "2021-01-02T20:04:47.000Z",
  "updated": "2021-01-02T20:04:47.000Z",
  "title": "Minimizing the number of complete bipartite graphs in a $K_s$-saturated graph",
  "authors": [
    "Beka Ergemlidze",
    "Abhishek Methuku",
    "Michael Tait",
    "Craig Timmons"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s \\geq 3$ and $t \\geq 2$, the minimum number of copies of $K_{1,t}$ in a $K_s$-saturated graph is $\\Theta ( n^{t/2})$. More precise results are obtained when $t = 2$ where the problem is related to Moore graphs with diameter 2 and girth 5. We prove that for $s \\geq 4$ and $t \\geq 3$, the minimum number of copies of $K_{2,t}$ in an $n$-vertex $K_s$-saturated graph is at least $\\Omega( n^{t/5 + 8/5})$ and at most $O(n^{t/2 + 3/2})$. These results answer a question of Chakraborti and Loh. General estimates on the number of copies of $K_{a,b}$ in a $K_s$-saturated graph are also obtained, but finding an asymptotic formula remains open.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-02T20:04:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "saturated graph",
      "complete bipartite graphs",
      "asymptotic formula remains open",
      "minimum number",
      "moore graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}