{
  "id": "2304.10353",
  "version": "v1",
  "published": "2023-04-20T14:49:07.000Z",
  "updated": "2023-04-20T14:49:07.000Z",
  "title": "Sparse vertex cutsets and the maximum degree",
  "authors": [
    "Stéphane Bessy",
    "Johannes Rauch",
    "Dieter Rautenbach",
    "Uéverton S. Souza"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that every graph $G$ of maximum degree $\\Delta$ and sufficiently large order has a vertex cutset $S$ of order at most $\\Delta$ that induces a subgraph $G[S]$ of maximum degree at most $\\Delta-3$. For $\\Delta\\in \\{ 4,5\\}$, we refine this result by considering also the average degree of $G[S]$. If $G$ has no $K_{r,r}$ subgraph, then we show the existence of a vertex cutset that induces a subgraph of maximum degree at most $\\left(1-\\frac{1}{{r\\choose 2}}\\right)\\Delta+O(1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-20T14:49:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximum degree",
      "sparse vertex cutsets",
      "average degree",
      "sufficiently large order"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}