{
  "id": "2009.03418",
  "version": "v1",
  "published": "2020-09-07T21:03:07.000Z",
  "updated": "2020-09-07T21:03:07.000Z",
  "title": "On a conjecture by Anthony Hill",
  "authors": [
    "Bojan Mohar"
  ],
  "comment": "6 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In the 1950's, English painter Anthony Hill described drawings of complete graphs $K_n$ in the plane having precisely $$H(n) = \\tfrac{1}{4}\\lfloor \\tfrac{n}{2}\\rfloor \\, \\lfloor \\tfrac{n-1}{2}\\rfloor \\, \\lfloor \\tfrac{n-2}{2}\\rfloor \\,\\lfloor \\tfrac{n-3}{2}\\rfloor$$ crossings. It became a conjecture that this number is minimum possible and, despite serious efforts, the conjecture is still widely open. Another way of drawing $K_n$ with the same number of crossings was found by Bla\\v{z}ek and Koman in 1963. In this note we provide, for the first time, a very general construction of drawings attaining the same bound. Surprisingly, the proof is extremely short and may as well qualify as a \"book proof\". In particular, it gives a very simple explanation of the phenomenon discovered by Moon in 1968 that a random set of $n$ points on the unit sphere $\\SS^2$ in $\\RR^3$ joined by geodesics gives rise to a drawing whose number of crossings asymptotically approaches the Hill value $H(n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-07T21:03:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "68R10"
    ],
    "keywords": [
      "conjecture",
      "english painter anthony hill",
      "hill value",
      "first time",
      "complete graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}