On Critical Circuits in k-Connected Matroids

Abstract

We show that, for every integer $k \ge 4$, if M is a k-connected matroid and C is a circuit of M such that for every $e \in C$, $M\backslash e$ is not k-connected, then C meets a cocircuit of size at most $2k -3$; furthermore, if M is binary and $k \ge 5$, then C meets a cocircuit of size at most $2k-4$. It follows from our results and a result of Reid et al  [5] that every minimally k-connected matroid has a cocircuit of size at most $2k-3$, and every minimally k-connected binary matroid has a cocircuit of size at most $2k-4$.