{
  "id": "1302.1386",
  "version": "v2",
  "published": "2013-02-06T14:35:45.000Z",
  "updated": "2013-11-06T22:41:05.000Z",
  "title": "Homometric sets in trees",
  "authors": [
    "Radoslav Fulek",
    "Slobodan Mitrović"
  ],
  "doi": "10.1016/j.ejc.2013.06.008",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G = (V,E)$ denote a simple graph with the vertex set $V$ and the edge set $E$. The profile of a vertex set $V'\\subseteq V$ denotes the multiset of pairwise distances between the vertices of $V'$. Two disjoint subsets of $V$ are \\emph{homometric}, if their profiles are the same. If $G$ is a tree on $n$ vertices we prove that its vertex sets contains a pair of disjoint homometric subsets of size at least $\\sqrt{n/2} - 1$. Previously it was known that such a pair of size at least roughly $n^{1/3}$ exists. We get a better result in case of haircomb trees, in which we are able to find a pair of disjoint homometric sets of size at least $cn^{2/3}$ for a constant $c > 0$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-06T22:41:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "disjoint homometric sets",
      "disjoint homometric subsets",
      "vertex sets contains",
      "haircomb trees",
      "disjoint subsets"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.1386F"
    }
  }
}