A matroid generalization of a result of Dirac

This paper generalizes a theorem of Dirac for graphs by proving that if M is a 3-connected matroid, then, for all pairs {a, b} of distinct elements of M and all cocircuits C* of M, there is a circuit that contains {a, b} and meets C*. It is also shown that, although the converse of this result fails, the specified condition call be used to characterize 3-connected matroids.