{
  "id": "2207.03747",
  "version": "v1",
  "published": "2022-07-08T08:32:13.000Z",
  "updated": "2022-07-08T08:32:13.000Z",
  "title": "On the size of matchings in 1-planar graph with high minimum degree",
  "authors": [
    "Yuanqiu Huang",
    "Zhangdong Ouyang",
    "Fengming Dong"
  ],
  "comment": "19 pages and 5 figures. To appear in SIAM Discrete Mathematics",
  "categories": [
    "math.CO"
  ],
  "abstract": "A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph with minimum degree 3 and $n\\geq 7$ vertices has a matching of size at least $\\frac{n+12}{7}$, which is tight for some graphs. They also provided tight lower bounds for the sizes of matchings in 1-planar graphs with minimum degree 4 or 5. In this paper, we show that any 1-planar graph with minimum degree 6 and $n \\geq 36$ vertices has a matching of size at least $\\frac{3n+4}{7}$, and this lower bound is tight. Our result confirms a conjecture posed by Biedl and Wittnebel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-08T08:32:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "05C35",
      "05C70"
    ],
    "keywords": [
      "high minimum degree",
      "tight lower bounds",
      "common end vertex",
      "result confirms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}