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 for x itself. Wild observed that every binary matroid M satisfies the following property: For all bases B and elements x, y of M , if R(x → B) = R(y → B), then x and y are either equal or parallel. He conjectured that this property characterizes binary matroids. We present a counterexample. Our counterexample, M , is simpler to describe via its dual M∗. Let M∗ be the matroid on {a, a′, b, b′, c, c′, d, d′} whose underlying simple matroid is the 4-point line (i.e., U2,4) and for which {a, a′}, {b, b′}, {c, c′}, and {d, d′} are parallel classes. (Thus M is simple and representable over every field other than GF (2).) The automorphism group of M is transitive on the bases, so it suffices to show that for a particular basis B of M and the two elements x, y of M not in B, R(x → B) and R(y → B) are unequal (thus proving the condition above for this non-binary matroid). Let B be the basis {a′, b′, c, c′, d, d′} of M . To find R(a → B), we find the cocircuit of M∗ contained in {a, a′, b′, c, c′, d, d′}, which is {a, a′, c, c′, d, d′}, and then delete a; thus R(a → B) = {a′, c, c′, d, d′}. Similarly R(b → B) = {b′, c, c′, d, d′}, proving that R(a→ B) and R(b→ B) are unequal, as needed. It is well-known that some basis-exchange properties characterize certain classes of matroids. To describe this efficiently, we adopt another notational convention from [9]. For bases B and B′ of a matroid M and an element x of B, let Sym(x, B,B′) be the set of elements y of B′ such that both (B−x)∪y and (B′−y)∪x are bases of M . Some basis-exchange properties discussed in [9] impose restrictions on the cardinality |Sym(x, B, B′)| of Sym(x, B, B′). The following proposition, conjectured by Rota and proven by Greene, was the first result of this type. (See [2, Section XI, Theorem 1]. For the motivation for studying this type of basis-exchange property, see [3] and [8].) Proposition 1. A matroid M is binary if and only if for each pair of bases B, B′ of M and each x ∈ B, |Sym(x, B, B′)| is odd. Greene showed the equivalence of several statements, all characterizing binary matroids, including the basis-exchange property in Proposition 1 and the following circuit-cocircuit intersection property from [4].