{
  "id": "1811.08334",
  "version": "v2",
  "published": "2018-11-20T16:03:01.000Z",
  "updated": "2020-06-17T13:58:23.000Z",
  "title": "An asymptotic resolution of a problem of Plesník",
  "authors": [
    "Stijn Cambie"
  ],
  "comment": "16 pages, 5 figures revised version as accepted at JCTB",
  "categories": [
    "math.CO"
  ],
  "abstract": "Fix $d \\ge 3$. We show the existence of a constant $c>0$ such that any graph of diameter at most $d$ has average distance at most $d-c \\frac{d^{3/2}}{\\sqrt n}$, where $n$ is the number of vertices. Moreover, we exhibit graphs certifying sharpness of this bound up to the choice of $c$. This constitutes an asymptotic solution to a longstanding open problem of Plesn\\'{i}k. Furthermore we solve the problem exactly for digraphs if the order is large compared with the diameter.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2020-06-17T13:58:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C20",
      "05C35"
    ],
    "keywords": [
      "asymptotic resolution",
      "longstanding open problem",
      "asymptotic solution",
      "graphs certifying sharpness",
      "average distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}