On the Length of a Partial Independent Transversal in a Matroidal Latin Square

TL;DR

This paper investigates a matroidal analogue of the Brualdi-Ryser conjecture, establishing bounds on the length of independent partial transversals in matrices with matroid bases, and demonstrating the tightness of these bounds.

Contribution

It introduces a matroidal version of the conjecture and proves bounds on the length of independent partial transversals in such matrices.

Findings

Any n×n matrix with matroid bases has an independent partial transversal of length at least ⌈2n/3⌉.

There exist matrices with maximal independent partial transversals of at most n-1.

The bounds are tight for certain matrices.

Abstract

We suggest and explore a matroidal version of the Brualdi - Ryser conjecture about Latin squares. We prove that any $n\times n$ matrix, whose rows and columns are bases of a matroid, has an independent partial transversal of length $\lceil2n/3\rceil$. We show that for any $n$, there exists such a matrix with a maximal independent partial transversal of length at most $n-1$.