### Abstract

Original language | English |
---|---|

Pages (from-to) | 1164 - 1171 |

Number of pages | 8 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 13 |

DOIs | |

Publication status | Published - 2011 |

### Cite this

*Discrete Mathematics*,

*311*(13), 1164 - 1171. https://doi.org/10.1016/j.disc.2010.06.026

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*Discrete Mathematics*, vol. 311, no. 13, pp. 1164 - 1171. https://doi.org/10.1016/j.disc.2010.06.026

**Multi-latin squares.** / Cavenagh, Nicholas; Hamalainen, Carlo; Lefevre, James; Stones, Douglas.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Multi-latin squares

AU - Cavenagh, Nicholas

AU - Hamalainen, Carlo

AU - Lefevre, James

AU - Stones, Douglas

PY - 2011

Y1 - 2011

N2 - A multi-latin square of order n and index k is an n x n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n >= 2m, thus generalizing Evans Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n >= k + 2. We also show that for each n >= 1, there exists some finite value g (n) such that for all k >= g (n), every k-latin square of order n is separable. We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders

AB - A multi-latin square of order n and index k is an n x n array of multisets, each of cardinality k, such that each symbol from a fixed set of size n occurs k times in each row and k times in each column. A multi-latin square of index k is also referred to as a k-latin square. A 1-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in k-latin square of order m embeds in a k-latin square of order n, for each n >= 2m, thus generalizing Evans Theorem. Exploiting this result, we show that there exist non-separable k-latin squares of order n for each n >= k + 2. We also show that for each n >= 1, there exists some finite value g (n) such that for all k >= g (n), every k-latin square of order n is separable. We discuss the connection between k-latin squares and related combinatorial objects such as orthogonal arrays, latin parallelepipeds, semi-latin squares and k-latin trades. We also enumerate and classify k-latin squares of small orders

UR - http://arxiv.org/abs/1007.4096

U2 - 10.1016/j.disc.2010.06.026

DO - 10.1016/j.disc.2010.06.026

M3 - Article

VL - 311

SP - 1164

EP - 1171

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 13

ER -