Cyclic orderings of matroids

The problem asks whether partitioning a matroid’s ground set into bases always yields a cyclic ordering whose every block of the prescribed length is a basis. The general conjecture remains open.

Known results

• Van den Heuvel and Thomassé proved the coprime case: if $\gcd(|E(M)|,r(M))=1$, the required ordering exists exactly under the relevant rank inequalities.
• Their result implies the problem whenever the ground set is partitioned into $k$ bases and $\gcd(k,r(M))=1$.
• The same work gives a block-level ordering in the general case, but ordering elements within blocks is unresolved; a counterexample shows the natural refinement need not work.

2024 split-matroid theorem

A 2024 paper claims that every split matroid whose ground set decomposes into pairwise disjoint bases is cyclically orderable, with a polynomial independence-oracle algorithm. This is substantial progress but does not settle the general problem; the reported theorem is treated here as unverified.

Current status (as of September 2026): The coprime and split-matroid cases are reported settled, while the general case remains open.