### Abstract

We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals. We prove the following: (1) The minimum size of a cover in a Latin square of order n is (Formula presented.) if and only if the maximum size of a partial transversal is either (Formula presented.) or (Formula presented.). (2) A minimal cover in a Latin square of order n has size at most (Formula presented.). (3) There are infinitely many orders n for which there exists a Latin square having a minimal cover of every size from n to (Formula presented.). (4) Every Latin square of order n has a minimal cover of a size which is asymptotically equal to (Formula presented.). (5) If (Formula presented.) and (Formula presented.) then there is a Latin square of order n with a maximal partial transversal of size (Formula presented.). (6) For any (Formula presented.), asymptotically almost all Latin squares have no maximal partial transversal of size less than (Formula presented.).

Original language | English |
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Pages (from-to) | 1109-1136 |

Number of pages | 28 |

Journal | Designs Codes and Cryptography |

Volume | 87 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2019 |

### Keywords

- Covers
- Latin square
- Transversal

### Cite this

*Designs Codes and Cryptography*,

*87*(5), 1109-1136. https://doi.org/10.1007/s10623-018-0499-9