Base orderable matroid

Brualdi showed that all matroids satisfy the following property: for any two bases $$B_1$$ and $$B_2$$ there is a bijection $$f: B_1 \to B_2$$ with the property that $$B_1-e+f(e)$$ is a base for any $$e \in B_1$$ .

A matroid is weakly base orderable if for any two bases $$B_1$$ and $$B_2$$ there is a bijection $$f: B_1 \to B_2$$ with the property that $$B_1-e+f(e)$$ and $$B_2-f(e)+e$$ are bases for any $$e \in B_1$$ .

A matroid is strongly base orderable if for any two bases $$B_1$$ and $$B_2$$ there is a bijection $$g: B_1 \to B_2$$ with the property that $$(B_1\setminus X)\cup g(X)$$ is a base for any $$X \subseteq B_1$$ .

Examples

The linear representation of the matroids $$P_7$$ (2 dim) and $$P_8$$ (3 dim)

• Every gammoid is strongly base orderable
• The matroid $$P_7$$ is strongly base orderable but not a gammoid (see page 128)
• The matroid $$P_8$$ is weakly base orderable but not strongly base orderable
• The cycle matroid of $$K_4$$ is not weakly base orderable

Properties

Davies and McDiarmid showed the following:

Theorem. If $$M_1=(S,{\mathcal F}_1)$$ and $$M_2=(S,{\mathcal F}_2)$$ are strongly base orderable matroids, and S can be partitioned into k independent subsets both in $$M_1$$ and in $$M_2$$ , then S can be partitioned into k common independent sets of $$M_1$$ and $$M_2$$ .