On basis-exchange properties for matroids

We give a counterexample to a conjecture by Wild about binary matroids. We connect two equivalent lines of research in matroid theory: a simple type of basis-exchange property and restrictions on the cardinalities of intersections of circuits and cocircuits. Finally, we characterize direct sums of series-parallel networks by a simple basis-exchange property. In [9], Wild proposed a characterization of binary matroids. In order to state his conjecture compactly, we recall his notation. (We also use standard matroid notation and terminology as found, for example, in [6].) For a basis B of a matroid M and an element x in the matroid, R(x → B) denotes the set of all elements y ∈ B so that (B − y) ∪ x is a basis of M. Informally, R(x → B) is the set of elements of B that can be replaced by x. Thus if x is in B, the set R(x → B) is the singleton {x}; if x is not in B, the set R(x → B) consists of all elements in the fundamental circuit, C(x, B), of x with respect to B, except f...

• Type: Journal
• Year: 1998
• Where: DM
• Authors: Joseph E. Bonin
• Keywords: Basis-exchange Property, Binary Matroids, DM 1998, Matroid