A generalized-polymatroid approach to disjoint common independent sets in two matroids

Abstract

In this paper, we investigate the classes of matroid intersection admitting a solution for the problem of partitioning the ground set E into k common independent sets, where E can be partitioned into k independent sets in each of the two matroids. For this problem, we present a new approach building upon the generalized-polymatroid intersection theorem. We exhibit that this approach offers alternative proofs and unified understanding of previous results showing that the problem has a solution for the intersection of two laminar matroids and that of two matroids without (k+1)-spanned elements. Moreover, we newly show that the intersection of a laminar matroid and a matroid without (k+1)-spanned elements admits a solution. We also construct an example of a transversal matroid which is incompatible with the generalized-polymatroid approach.