The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture

The Kajitani–Ueno–Miyano conjecture asserts that every finite uniformly dense matroid has a cyclic basis ordering. The conjecture is proved for all matroids of rank three. The new result establishes the previously unresolved divisible case, where the ground-set size is a multiple of three, without assumptions of simplicity, representability or paving. Together with the previously published coprime-case theorem of van den Heuvel and Thomassé, this covers every finite uniformly dense rank-three matroid. The unrestricted conjecture remains open in higher rank.

Proves the divisible rank-three case: rank exactly three and ground-set size a multiple of three, with no restriction to simple, paving, representable or graphic matroids. The part not previously in the literature is the non-simple sub-case, since McGuinness had settled all paving matroids and a rank-three matroid is paving exactly when it has no parallel pairs. Combined with the coprime-case theorem of van den Heuvel and Thomasse, this covers every finite uniformly dense rank-three matroid. The conjecture remains open in higher rank.