Some Properties on the Bicolor Basis Graph of A Matroid

Let φ be a function which is defined on elements of a matroid m onto f1; 2g. the bicolor basis graphg(b(m); φ), is a graph with vertex set given by the set of bases of m,b(m), in which two bases b1 andb2 are adjacent if b1 4 b2 = fe; fg and φ(e) 6= φ(f), where e 2 b1 and f 2 b2. in this article, wecharacterize some conditions on φ and m, such that they guarantee the graph g(b(m); φ) has a cycle orhas no pendant vertex.