Vertex disjoint cycles in intersection graphs of bases of matroids

The intersection graph of bases of a matroid M=(E, B) is a graph G=G I (M) with vertex set V(G) and edge set E(G) such that V(G)=B(M) and E(G)={BB′: |B ∩ B′|≠0, B, B′∈ B(M), where the same notation is used for the vertices of G and the bases of M. Suppose that |V (G I (M))| =n and k1 + k2... kp = n, where ki is an integer, i =1, 2,..., p. In this paper, we prove that there is a partition of V (G I (M)) into p parts V1, V2,..., Vp such that |Vi| = ki and the subgraph Hi induced by Vi contains a ki-cycle when ki ≥3, Hi is isomorphic to K2 when ki =2 and Hi is a single point when ki =1.