The Euler circuit theorem for binary matroids

It is proved that, if M is a binary matroid, then every cocircuit of M has even cardinality if and only if M can be obtained by contracting some other binary matroid M⁺ onto a single circuit. This is the natural analog of the Euler circuit theorem for graphs. It is also proved that every coloop-free matroid can be obtained by contracting some other matroid (not in general binary) onto a single circuit.