### Abstract

We prove that for all odd m >= 3 there exists a latin square of order 3 m that contains an (m - 1) x m latin subrectangle consisting of entries not in any transversal. We prove that for all even n >= 10 there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders n >= 11. Finally, we report on an extensive computational study of transversal-free entries and sets of disjoint transversals in the latin squares of order n

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

Pages (from-to) | 124 - 141 |

Number of pages | 18 |

Journal | Journal of Combinatorial Designs |

Volume | 20 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 |

## Cite this

Egan, J., & Wanless, I. M. (2012). Latin squares with restricted transversals.

*Journal of Combinatorial Designs*,*20*(2), 124 - 141. https://doi.org/10.1002/jcd.20297