On connectivity in matroids and graphs

Abstract

In this paper we derive several results for connected matroids and use these to obtain new results for -connected graphs. In particular, we generalize work of Murty and Seymour on the number of two-element cocircuits in a minimally connected matroid, and work of Dirac, Plummer and Mader on the number of vertices of degree two in a minimally $2$-connected graph. We also solve a problem of Murty by giving a straightforward but useful characterization of minimally connected matroids. The final part of the paper gives a matroid generalization of Dirac and Plummer’s result that every minimally $2$-connected graph is $3$-colourable.