An Algorithmic Proof of the Piff–Welsh Theorem on Transversal Matroid Representations

On the automorphism group of a matroid

Transversal and Cotransversal Matroids via their Representations

It is known that the duals of transversal matroids are precisely the strict gammoids. We show that, by representing these two families of matroids geometrically, one obtains a simple proof of their duality.

On ternary transversal matroids

Ingleton [8, p. 123] raised the question of characterizing the class of transversal matroids that are representable over some particular field F. He noted that when F = GF(2), this problem had already been solved by de Sousa and Welsh [6] who showed that the class of binary transversal matroids coincides with the class of graphic transversal matroids. The latter class had earlier been character...

On the critical exponent of transversal matroids

Brylawski [2] has shown that loopless principal transversal matroids have critical exponent at most 2. Welsh [4] asks if a similar result holds for all transversal matroids. We answer in the affirmative by proving that all loopless transversal matroids have critical exponent at most 2. The terminology used here for matroids will in general follow Welsh 131. A rank r transversal matroid M is rep...

Landau's and Rado's Theorems and Partial Tournaments

Using Rado’s theorem for the existence of an independent transversal of family of subsets of a set on which a matroid is defined, we give a proof of Landau’s theorem for the existence of a tournament with a prescribed degree sequence. A similar approach is used to determine when a partial tournament can be extended to a tournament with a prescribed degree sequence. Mathematics Subject Classific...