On fundamental transversal matroids

Unified proofs of two theorems on fundamental transversal matroids are presented. A necessary condition for a matroid to be a fundamental transversal matroid with respect to a given basis is given. The purpose of this note is to prove a theorem about fundamental transversal matroids which in turn yields unified proofs of two recent theorems about such matroids. We assume the reader has a modest acquaintance with the basic notions of matroid theory. Thus he or she should know that a matroid (or combinatorial pregeometry) M on a finite set E consists of a nonempty, hereditary collection of subsets of E which are called independent sets, and that the maximal independent sets all have the same cardinality (the rank of M) and are called bases. Corresponding to a matroid M there • are circuits (minimal nonindependent sets) and a dual matroid M whose bases (cobases of M) are the complements of the bases of M. The circuits of M are called cocircuits (or bonds) of M. A property important in what follows is that a cocircuit and circuit cannot have exactly one element in common. A matroid is nonseparable if every pair of elements of E lie in a common circuit; otherwise there is a partition E . (l < i < t) of E and matroids M. on E . (l < i < t) such that M is the direct sum M, © • ■ • © M . For more details one can consult [2], [4], or [7]. An interesting kind of matroid is a transversal matroid [3] whose independent sets are the partial transversals of a family of sets. If (A .: i £ I) is a family of subsets of set E, then T(A.: i e /) denotes the corresponding transversal matroid. Suppose M is a matroid on E and ß is a basis of M. Each e e B gives rise to a unique cocircuit C* with e e C* C (E\b) u\e\ called the fundamental cocircuit of e with respect to the cobasis E\B. For e £ E\B Received by the editors October 12, 1972 and, in revised form, June 22, 1973. AMS (MOS) subject classifications (1970). Primary 05B35; Secondary 05A05, 05C99. Research supported in part by National Science Foundation Grant No. GP17815. Copyright © 1974, American Mathematical Society 151 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use