{
  "id": "1908.05072",
  "version": "v1",
  "published": "2019-08-14T11:14:49.000Z",
  "updated": "2019-08-14T11:14:49.000Z",
  "title": "Light edges in 1-planar graphs of minimum degree 3",
  "authors": [
    "Bei Niu",
    "Xin Zhang"
  ],
  "comment": "This paper was submitted to Discrete Mathematics on Dec.4, 2018, and will be published there",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one another edge. In this work we prove that each 1-planar graph of minimum degree at least $3$ contains an edge with degrees of its endvertices of type $(3,\\leq23)$ or $(4,\\leq11)$ or $(5,\\leq9)$ or $(6,\\leq8)$ or $(7,7)$. Moreover, the upper bounds $9,8$ and $7$ here are sharp and the upper bounds $23$ and $11$ are very close to the possible sharp ones, which may be 20 and 10, respectively. This generalizes a result of Fabrici and Madaras [Discrete Math., 307 (2007) 854--865] which says that each 3-connected 1-planar graph contains a light edge, and improves a result of Hud\\'ak and \\v{S}ugerek [Discuss. Math. Graph Theory, 32(3) (2012) 545--556], which states that each 1-planar graph of minimum degree at least $4$ contains an edge with degrees of its endvertices of type $(4,\\leq 13)$ or $(5,\\leq 9)$ or $(6,\\leq 8)$ or $(7, 7)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-14T11:14:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimum degree",
      "light edge",
      "upper bounds",
      "endvertices",
      "graph theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}