Complement minimally non-totally unimodular matrices

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Abstract

We prove that, up to row and column permutations and complement operations, the only complement minimally non-totally unimodular matrices are the cycle matrices $C_3$ and $C_5$. This settles a conjecture of Chervet, Grappe, and Vallée. As a consequence, every simplicial cone generated by the rows of a totally equimodular matrix admits a regular unimodular Hilbert triangulation.

Citation

Choi, Suyoung, and Mathieu Vallée. 2026. “Complement minimally non-totally unimodular matrices.” https://arxiv.org/abs/2608.08334

@misc{choi_complement_2026,
  author = {Choi, Suyoung and Vallée, Mathieu},
  year = {2026},
  title = {Complement minimally non-totally unimodular matrices},
  doi = {10.48550/arXiv.2608.08334},
  url = {http://arxiv.org/abs/2608.08334},
  note = {arXiv:2608.08334 [math.CO]},
  publisher = {arXiv},
}