Gabow’s full serial decomposition conjecture

Prove Gabow’s conjecture that, for two matroid bases A and B, the exchange decomposition can be chosen with one successive exchange layer for each element of A\setminus B, equivalently with k=|A\setminus B|-1 in Gabow’s chain formulation.

Background

Gabow’s conjecture is a special, particularly strong case of the serial-exchange program. It asks for a complete chain decomposition of exchanges between two bases, with the maximum possible number of nontrivial stages.

The paper proves only relaxed chain statements with logarithmic-length guarantees and explains that the full value of k remains conjectural. A proof would have implications for matroid-constrained fair division and conjectures concerning generators of toric ideals.

References

The statement usually referred to as Gabow's conjecture is his question whether $k$ can be chosen to be $|A\setminus B|-1$ if $X=A\setminus B$ and $Y=B\setminus A$.

— Generalizing the Multiple Exchange Property for Matroid Bases  (2511.16021 - Oki et al., 20 Nov 2025) in Remark 2.??, Section 2.3, paragraph “Relaxed Gabow’s conjecture and base orderability”