{
  "id": "2105.00980",
  "version": "v1",
  "published": "2021-05-03T16:31:15.000Z",
  "updated": "2021-05-03T16:31:15.000Z",
  "title": "Growable Realizations: a Powerful Approach to the Buratti-Horak-Rosa Conjecture",
  "authors": [
    "M. A. Ollis",
    "Anita Pasotti",
    "Marco A. Pellegrini",
    "John R. Schmitt"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Label the vertices of the complete graph $K_v$ with the integers $\\{ 0, 1, \\ldots, v-1 \\}$ and define the length of the edge between $x$ and $y$ to be $\\min( |x-y| , v - |x-y| )$. Let $L$ be a multiset of size $v-1$ with underlying set contained in $\\{ 1, \\ldots, \\lfloor v/2 \\rfloor \\}$. The Buratti-Horak-Rosa Conjecture is that there is a Hamiltonian path in $K_v$ whose edge lengths are exactly $L$ if and only if for any divisor $d$ of $v$ the number of multiples of $d$ appearing in $L$ is at most $v-d$. We introduce \"growable realizations,\" which enable us to prove many new instances of the conjecture and to reprove known results in a simpler way. As examples of the new method, we give a complete solution when the underlying set is contained in $\\{ 1,4,5 \\}$ or in $\\{ 1,2,3,4 \\}$ and a partial result when the underlying set has the form $\\{ 1, x, 2x \\}$. We believe that for any set $U$ of positive integers there is a finite set of growable realizations that implies the truth of the Buratti-Horak-Rosa Conjecture for all but finitely many multisets with underlying set $U$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-03T16:31:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C38",
      "05C78"
    ],
    "keywords": [
      "buratti-horak-rosa conjecture",
      "growable realizations",
      "powerful approach",
      "finite set",
      "complete graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}