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.