A Fano framework for binary delta-matroids
Abstract
Dunshee and Ellingham recently showed that seven natural properties of a cellularly embedded graph form a Fano-plane framework. We establish an analogous framework for binary delta-matroids. For a binary delta-matroid D and τ, let Z3(D,τ) denote its associated binary tight $3$-matroid. The six outer points are represented by evenness or bipartiteness of D and its global vertex-flip transforms. For the seventh point, we call D Z3-bipartite when every circuit of Z3(D,τ) has even cardinality. We show that the satisfied properties are precisely the nonzero vectors of a subspace of Ftwo3. For ribbon-graphic delta-matroids, Z3-bipartiteness is equivalent to bipartiteness of the medial graph, so the construction recovers the Fano-plane framework for embedded graphs.
Related
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- Valuated delta matroids and principal minors of Hermitian matrices
- The Miyaoka-Yau inequality and the delta invariant for Fano varieties I -klt Fano case-
- Oriented and Valuated Delta Matroids from Stable Polynomials
- The Miyaoka-Yau inequality and the delta invariant for Fano varieties II -soliton case-
BibTeX
@misc{li2026fano,
title = {{A Fano framework for binary delta-matroids}},
author = {Zhuo Li and Xian'an Jin and Qi Yan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08588},
eprint = {2609.08588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08588},
}
RIS
TY - UNPB
TI - A Fano framework for binary delta-matroids
AU - Zhuo Li
AU - Xian'an Jin
AU - Qi Yan
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08588
UR - https://arxiv.org/abs/2609.08588
AB - Dunshee and Ellingham recently showed that seven natural properties of a cellularly embedded graph form a Fano-plane framework. We establish an analogous framework for binary delta-matroids. For a binary delta-matroid D and τ, let Z_3(D,τ) denote its associated binary tight $3$-matroid. The six outer points are represented by evenness or bipartiteness of D and its global vertex-flip transforms. For the seventh point, we call D Z_3-bipartite when every circuit of Z_3(D,τ) has even cardinality. We show that the satisfied properties are precisely the nonzero vectors of a subspace of Ftwo^3. For ribbon-graphic delta-matroids, Z_3-bipartiteness is equivalent to bipartiteness of the medial graph, so the construction recovers the Fano-plane framework for embedded graphs.
ER -