{
  "id": "2102.10916",
  "version": "v1",
  "published": "2021-02-22T11:36:45.000Z",
  "updated": "2021-02-22T11:36:45.000Z",
  "title": "On Metric Dimensions of Hypercubes",
  "authors": [
    "Aleksander Kelenc",
    "Aoden Teo Masa Toshi",
    "Riste Skrekovski",
    "Ismael G. Yero"
  ],
  "comment": "9 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First, we show that the metric and edge metric dimension of $Q_d$ differ by only one for every integer $d$. In particular, if $d$ is odd, then the metric and edge metric dimensions of $Q_d$ are equal. Second, we prove that the metric and mixed metric dimensions of the hypercube $Q_d$ are equal for every $d \\ge 3$. We conclude the paper by conjecturing that all these three types of metric dimensions of $Q_d$ are equal when $d$ is large enough.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-22T11:36:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C76"
    ],
    "keywords": [
      "edge metric dimension",
      "distinct vertices",
      "hypercube graphs",
      "smallest ordered set",
      "mixed metric dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}