On the basis graph of a bicolored matroid

The basis graph of a matroid M is the graph G(B(M)) whose vertex set is the set of basis of M and two basis B and B′ are adjacent if the size of its symmetric difference is two. We say that ϕ:E(M)→{1,2} is an effective 2-coloring of M if ϕ is a surjective function. Let M be a matroid and ϕ and effective 2-coloring of M, we define the bicolor basis graph, G(B(M),ϕ) with respet to ϕ of a matroid M as the spanning subgraph of G(B(M)) such that two basis B and B′ are adjacent if they are adjacent in G(B(M)) and the restriction of ϕ to their symmetric difference is effective. Our main result states that if M is a connected matroid and ϕ is an effective 2-coloring of M, then G(B(M),ϕ) is connected.