{
  "id": "2306.08217",
  "version": "v1",
  "published": "2023-06-14T03:11:55.000Z",
  "updated": "2023-06-14T03:11:55.000Z",
  "title": "Partitioning graphs with linear minimum degree",
  "authors": [
    "Jie Ma",
    "Hehui Wu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that there exists an absolute constant $C>0$ such that, for any positive integer $k$, every graph $G$ with minimum degree at least $Ck$ admits a vertex-partition $V(G)=S\\cup T$, where both $G[S]$ and $G[T]$ have minimum degree at least $k$, and every vertex in $S$ has at least $k$ neighbors in $T$. This confirms a question posted by K\\\"uhn and Osthus and is tight up to a constant factor. Our proof combines probabilistic methods with structural arguments based on Ore's Theorem on $f$-factors of bipartite graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-14T03:11:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear minimum degree",
      "partitioning graphs",
      "absolute constant",
      "structural arguments",
      "probabilistic methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}