Enumerating toric-colorable seeds of Picard number five via binary matroids

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Abstract

We introduce a binary matroid framework for the enumeration of mod $2$ toric-colorable seeds of fixed Picard number. Working with binary matroids up to isomorphism, we organize them through their contraction structure and recursively enumerate weak pseudomanifold subcomplexes by a dynamic programming algorithm. The resulting method combines these structural reductions with a Gray-code traversal of the mod $2$ kernel of the ridge–facet incidence matrix. Using this framework, we complete the classification of mod $2$ toric-colorable seeds of dimension four and Picard number five, proving that there are exactly $198,846$ isomorphism classes. This case was computationally infeasible for the GPU-based algorithm previously used by Choi, Jang, and Vallée to treat Picard number four. We further verify that each of these seeds admits an integral characteristic map. As a validation of the method, we also reproduce that Picard number four classification: the weak pseudomanifold enumeration stage drops from over ten days on a GPU to ten minutes on a single CPU.

Citation

Choi, Suyoung, and Mathieu Vallée. 2026. “Enumerating toric-colorable seeds of Picard number five via binary matroids.” https://arxiv.org/abs/2606.29309

@misc{choi_enumerating_2026,
  author = {Choi, Suyoung and Vallée, Mathieu},
  year = {2026},
  title = {Enumerating toric-colorable seeds of Picard number five via binary matroids},
  doi = {10.48550/arXiv.2606.29309},
  url = {http://arxiv.org/abs/2606.29309},
  note = {arXiv:2606.29309 [math.CO]},
  publisher = {arXiv},
}