A Matroid Generalization of a Result on Row-Latin Rectangles

摘要

Let A be an m * n matrix in which the entries of each row are all distinct. A. A. Drisko (1998, J. Combin. Theory Ser. A 84, 181-195) showed that if m >= 2n - 1, then A has a transversal: a set of n distinct entries with no two in the same row or column. We generalize this to matrices with entries in the ground set of a matroid. For such a matrix A, we show that if each row of A forms an independent set, then we can require the transversal to be independent as well. We determine the complexity of an algorithm based on the proof of this result. Finally, we observe that m >= 2n - 1 appears to force the existence of not merely one but many transversals. We discuss a number of conjectures related to this observation (some of which involve matroids and some of which do not).