{
  "id": "1912.07377",
  "version": "v1",
  "published": "2019-12-13T13:11:31.000Z",
  "updated": "2019-12-13T13:11:31.000Z",
  "title": "Further progress on the Buratti-Horak-Rosa conjecture",
  "authors": [
    "Anita Pasotti",
    "Marco Antonio Pellegrini"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The conjecture posed by Marco Buratti, Peter Horak and Alex Rosa states that a multiset $L$ of $v-1$ positive integers not exceeding $\\left\\lfloor \\frac{v}{2}\\right\\rfloor$ is the list of \\emph{edge-lengths} of a suitable Hamiltonian path of the complete graph with vertex-set $\\{0,1,\\ldots,v-1\\}$ if and only if, for every divisor $d$ of $v$, the number of multiples of $d$ appearing in $L$ is at most $v-d$. Here, we prove the validity of this conjecture for the lists $\\{1^a,x^b,(x+1)^c\\}$, with $x\\geq 3$ and $a,b,c\\geq 0$, in each of the following cases: (i) $x$ is odd and $a\\geq \\min\\{3x-3, b+2x-3 \\}$; (ii) $x$ is odd, $a\\geq 2x-2$ and $c\\geq \\frac{4}{3} b$; (iii) $x$ is even and $a\\geq \\min\\{3x-1, c+2x-1\\}$; (iv) $x$ is even, $a\\geq 2x-1$ and $b\\geq c$. As a consequence we also obtain some results on lists of the form $\\{y^a,z^b,(y+z)^c\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-13T13:11:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C38"
    ],
    "keywords": [
      "buratti-horak-rosa conjecture",
      "alex rosa states",
      "peter horak",
      "marco buratti",
      "suitable hamiltonian path"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}