{
  "id": "1912.11230",
  "version": "v1",
  "published": "2019-12-24T07:02:26.000Z",
  "updated": "2019-12-24T07:02:26.000Z",
  "title": "Parity of transversals of Latin squares",
  "authors": [
    "Darcy Best",
    "Ian M. Wanless"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \\bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\\equiv t_{cd}\\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} \\equiv 0 \\bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\\equiv0\\bmod 4$. We also show that $${\\rm per}\\, A(a \\mid c)+{\\rm per}\\, A(b \\mid c)+{\\rm per}\\, A(a \\mid d)+{\\rm per}\\, A(b \\mid d) \\equiv 0 \\bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\\equiv2\\bmod4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-24T07:02:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "transversals",
      "odd order",
      "columns sums equal",
      "regular bipartite graph",
      "perfect matchings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}