{
  "id": "1605.07613",
  "version": "v1",
  "published": "2016-05-24T17:43:41.000Z",
  "updated": "2016-05-24T17:43:41.000Z",
  "title": "The square of the 9-hypercube is 14-colorable",
  "authors": [
    "Juho Lauri"
  ],
  "comment": "3 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The $n$-hypercube, denoted by $Q_n$, has a vertex for each bit string of length $n$ with two vertices adjacent whenever their Hamming distance is one. The minimum number of colors needed to color $Q_n$ such that no two vertices at a distance at most $k$ receive the same color is denoted by $\\chi_{\\bar{k}}(n)$. Equivalently, $\\chi_{\\bar{k}}(n)$ denotes the minimum number of binary codes with minimum distance at least $k+1$ required to partition the $n$-dimensional Hamming space. Using a computer search, we improve upon the known upper bound for $n=9$ by showing that $13 \\leq \\chi_{\\bar{2}}(9) \\leq 14$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-24T17:43:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "minimum number",
      "vertices adjacent",
      "upper bound",
      "binary codes",
      "minimum distance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}