{
  "id": "math/0509410",
  "version": "v1",
  "published": "2005-09-19T00:48:42.000Z",
  "updated": "2005-09-19T00:48:42.000Z",
  "title": "Latin squares and their defining sets",
  "authors": [
    "Karola Meszaros"
  ],
  "comment": "16 pages, 24 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A Latin square $L(n,k)$ is a square of order $n$ with its entries colored with $k$ colors so that all the entries in a row or column have different colors. Let $d(L(n,k))$ be the minimal number of colored entries of an $n \\times n$ square such that there is a unique way of coloring of the yet uncolored entries in order to obtain a Latin square $L(n, k)$. In this paper we discuss the properties of $d(L(n,k))$ for $k=2n-1$ and $k=2n-2$. We give an alternate proof of the identity $d(L(n, 2n-1))=n^2-n$, which holds for even $n$, and we establish the new result $d(L(n, 2n-2)) \\geq n^2-\\lfloor\\frac{8n}{5}\\rfloor$ and show that this bound is tight for $n$ divisible by 10.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-09-19T00:48:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B15"
    ],
    "keywords": [
      "latin square",
      "defining sets",
      "minimal number",
      "unique way",
      "alternate proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9410M"
    }
  }
}