The basis graph of a bicolored matroid ?
Let ? ? be a 2-coloring of the elements of a matroid M M . The bicolor basis graph  of M M is the graph G(B(M),?) G ( B ( M ) , ? ) with vertex set given by the set of bases of M M in which two bases B B and B? B ? are adjacent if B?=(B?e)?f B ? = ( B ? e ) ? f for some elements e?B e ? B and f?B? f ? B ? with ?(e)??(f) ? ( e ) ? ? ( f ) . Let M M be a matroid with at least one circuit, we prove that G(B(M),?) G ( B ( M ) , ? ) is connected for every 2-coloring ? ? of M M if and only if M M is a connected matroid.
basis graph tree graph