Small Latin arrays have a near transversal

TL;DR

This paper computationally demonstrates that Latin arrays of order up to 11 always contain a diagonal with nearly all distinct symbols, implying the existence of near transversals, and extends bounds for larger arrays.

Contribution

It provides the first computational verification of near transversals in Latin arrays of small order and establishes bounds for larger arrays without such diagonals.

Findings

Latin arrays of order ≤11 have diagonals with at least n-1 distinct symbols

Existence of near transversals in Latin squares of these orders is confirmed

Lower bounds are computed for arrays lacking diagonals with many distinct symbols

Abstract

A Latin array is a matrix of symbols in which no symbol occurs more than once within a row or within a column. A diagonal of an $n\times n$ array is a selection of $n$ cells taken from different rows and columns of the array. The weight of a diagonal is the number of different symbols on it. We show via computation that every Latin array of order $n\le11$ has a diagonal of weight at least $n-1$. A corollary is the existence of near transversals in Latin squares of these orders. More generally, for all $k\le20$ we compute a lower bound on the order of any Latin array that does not have a diagonal of weight at least $n-k$.