Vertex disjoint cycles in intersection graphs of bases of matroids

Abstract

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... k_p = n, where k_i 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,..., V_p such that |V_i| = k_i and the subgraph H_i induced by V_i contains a k_i-cycle when k_i ≥3, H_i is isomorphic to K₂ when k_i =2 and H_i is a single point when k_i =1.