Transversals in Latin arrays with many distinct symbols

TL;DR

This paper proves that Latin arrays with a high number of distinct symbols always contain a transversal, and it characterizes smaller Latin arrays without transversals through computational methods.

Contribution

It establishes new bounds on the number of distinct symbols guaranteeing the existence of transversals in Latin arrays and classifies small Latin arrays lacking transversals.

Findings

Latin arrays with at least $(2-rac{1}{ oot{2}}) n^2$ symbols have transversals.

Row-Latin arrays with at least $rac{1}{4}(5- oot{5}) n^2$ symbols have transversals.

All Latin arrays of order 7 have transversals; smaller non-transversal arrays are classified.

Abstract

An array is row-Latin if no symbol is repeated within any row. An array is Latin if it and its transpose are both row-Latin. A transversal in an $n\times n$ array is a selection of $n$ different symbols from different rows and different columns. We prove that every $n \times n$ Latin array containing at least $(2-\sqrt{2}) n^2$ distinct symbols has a transversal. Also, every $n \times n$ row-Latin array containing at least $\frac14(5-\sqrt{5})n^2$ distinct symbols has a transversal. Finally, we show by computation that every Latin array of order $7$ has a transversal, and we describe all smaller Latin arrays that have no transversal.