{
  "id": "2310.15650",
  "version": "v1",
  "published": "2023-10-24T09:09:00.000Z",
  "updated": "2023-10-24T09:09:00.000Z",
  "title": "A characterization on orientations of graphs avoiding given lists on out-degrees",
  "authors": [
    "Xinxin Ma",
    "Hongliang Lu"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph and $F:V(G)\\to2^N$ be a set function. The graph $G$ is said to be \\emph{F-avoiding} if there exists an orientation $O$ of $G$ such that $d^+_O(v)\\notin F(v)$ for every $v\\in V(G)$, where $d^+_O(v)$ denotes the out-degree of $v$ in the directed graph $G$ with respect to $O$. In this paper, we give a Tutte-type good characterization to decide the $F$-avoiding problem when for every $v\\in V(G)$, $|F(v)|\\leq \\frac{1}{2}(d_G(v)+1)$ and $F(v)$ contains no two consecutive integers. Our proof also gives a simple polynomial algorithm to find a desired orientation. As a corollary, we prove the following result: if for every $v\\in V(G)$, $|F(v)|\\leq \\frac{1}{2}(d_G(v)+1)$ and $F(v)$ contains no two consecutive integers, then $G$ is $F$-avoiding. This partly answers a problem proposed by Akbari et. al.(2020)",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-24T09:09:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "orientation",
      "graphs avoiding",
      "characterization",
      "out-degree",
      "simple polynomial algorithm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}