Characterizations of transversal and fundamental transversal matroids

TL;DR

This paper characterizes fundamental transversal matroids by extending Mason's inequalities, showing they satisfy equality in these inequalities, and simplifies the understanding of their structure compared to general transversal matroids.

Contribution

It provides new characterizations of fundamental transversal matroids, including their relation to Mason's inequalities and a simpler proof of Brylawski's characterization.

Findings

Fundamental transversal matroids satisfy equality in Mason's inequalities.

They are characterized by specific rank conditions on cyclic flats.

The paper simplifies the understanding of fundamental transversal matroids.

Abstract

A result of Mason, as refined by Ingleton, characterizes transversal matroids as the matroids that satisfy a set of inequalities that relate the ranks of intersections and unions of nonempty sets of cyclic flats. We prove counterparts, for fundamental transversal matroids, of this and other characterizations of transversal matroids. In particular, we show that fundamental transversal matroids are precisely the matroids that yield equality in Mason's inequalities and we deduce a characterization of fundamental transversal matroids due to Brylawski from this simpler characterization.