On independent transversals in matroidal Latin rectangles

Abstract

A Latin rectangle is an m × n matrix in which the entries of each row and each column are all distinct numbers. In a row-Latin rectangle only the rows are required to consist of distinct entries. A partial transversal in a Latin rectangle is a set of distinct numbers such that no two of them are in the same row or column. A well-known conjecture of Ryser, Brualdi and Stein [2,8] asserts that every n × n Latin rectangle has a partial transversal of size n − 1. Woolbright [9], and independently and Brouwer, de Vries and Wieringa [1], proved the existence of a partial transversal of size $n - \sqrt n$ and Shor and Hatami [5,7] improved the bound to n – O (log² n).