The e-Exchange Basis Graph and Matroid connectedness

Abstract Let M be a matroid and e ∈ E ( M ) . The e -exchange basis graph of M has vertices labeled by bases of M , and two vertices are adjacent when the bases labeling them have symmetric difference { e , x } for some x ∈ E ( M ) . In this paper we show that a connected matroid is exactly a matroid with the property that for every element e ∈ E ( M ) , the e -exchange basis graph is connected.