# Base orderable matroid

From Egres Open

Brualdi ^{[1]} showed that all matroids satisfy the following property: for any two bases and there is a bijection with the property that is a base for any .

A matroid is **weakly base orderable** if for any two bases and there is a bijection with the property that and are bases for any .

A matroid is **strongly base orderable** if for any two bases and there is a bijection with the property that is a base for any .

## Examples

- Every gammoid is strongly base orderable
- The matroid is strongly base orderable but not a gammoid (see
^{[2]}page 128) - The matroid is weakly base orderable but not strongly base orderable
- The cycle matroid of is not weakly base orderable

## Properties

Davies and McDiarmid ^{[3]} showed the following:

**Theorem.** If and are strongly base orderable matroids, and *S* can be partitioned into *k* independent subsets both in and in , then *S* can be partitioned into *k* common independent sets of and .

## References

- ↑ Brualdi, R.A.,
*Comments on bases in dependence structures*, Bull. of the Australian Math. Soc. 1 (1969), 161–167, DOI link - ↑ A.W. Ingleton,
*Transversal matroids and related structures*, DOI link - ↑ J. Davies and C. Mcdiarmid,
*Disjoint common transversals and exchange structures*, J. London Math. Soc. 14 (1976), 55–62.DOI link.