{
  "id": "2402.05077",
  "version": "v1",
  "published": "2024-02-07T18:31:06.000Z",
  "updated": "2024-02-07T18:31:06.000Z",
  "title": "Cycle-factors in oriented graphs",
  "authors": [
    "Zhilan Wang",
    "Jin Yan",
    "Jie Zhang"
  ],
  "comment": "27 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $k$ be a positive integer. A $k$-cycle-factor of an oriented graph is a set of disjoint cycles of length $k$ that covers all vertices of the graph. In this paper, we prove that there exists a positive constant $c$ such that for $n$ sufficiently large, any oriented graph on $n$ vertices with both minimum out-degree and minimum in-degree at least $(1/2-c)n$ contains a $k$-cycle-factor for any $k\\geq4$. Additionally, under the same hypotheses, we also show that for any sequence $n_1, \\ldots, n_t$ with $\\sum^t_{i=1}n_i=n$ and the number of the $n_i$ equal to $3$ is $\\alpha n$, where $\\alpha$ is any real number with $0<\\alpha<1/3$, the oriented graph $D$ contains $t$ disjoint cycles of lengths $n_1, \\ldots, n_t$. This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-07T18:31:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C70",
      "05C20",
      "05C38"
    ],
    "keywords": [
      "oriented graph",
      "cycle-factor",
      "disjoint cycles",
      "minimum out-degree",
      "minimum in-degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}