A Note on a Nearly Uniform Partition into Common Independent Sets of Two Matroids

TL;DR

This paper strengthens previous results by showing that, under certain conditions, the ground set of two matroids can be partitioned into nearly uniform common independent sets, with set sizes differing by at most one.

Contribution

It proves the existence of a nearly uniform partition into common independent sets of two matroids, refining earlier results on such partitions.

Findings

Existence of nearly uniform partitions into common independent sets

Cardinality difference between sets is at most one

Builds on generalized-polymatroid approach

Abstract

The present note is a strengthening of a recent paper by K. Takazawa and Y. Yokoi (A generalized-polymatroid approach to disjoint common independent sets in two matroids, Discrete Mathematics (2019)). For given two matroids on $E$, under the same assumption in their paper to guarantee the existence of a partition of $E$ into $k$ common independent sets of the two matroids, we show that there exists a nearly uniform partition $\mathcal{P}$ of $E$ into $k$ common independent sets, where the difference of the cardinalities of any two sets in $\mathcal{P}$ is at most one.