发表机构
School of Mathematical Sciences, Xiamen University; School of Mathematics and Statistics, Lanzhou University; School of Mathematics and Statistics, Qinghai Minzu University（厦门大学数学科学学院; 兰州大学数学与统计学院; 青海民族大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为二元delta-拟阵建立了类似嵌入图的Fano平面框架，通过$Z_3$-二部性等性质刻画满足条件的非零向量，并恢复已知结果。
AI 中文摘要
Dunshee和Ellingham最近证明了细胞嵌入图的七个自然性质构成一个Fano平面框架。我们为二元delta-拟阵建立了类似的框架。对于二元delta-拟阵$D$和$\tau$，令$Z_3(D,\tau)$表示其关联的二元紧$3$-拟阵。六个外点由$D$及其全局顶点翻转变换的偶性或二部性表示。对于第七个点，当$Z_3(D,\tau)$的每个回路具有偶数基数时，我们称$D$为$Z_3$-二部的。我们证明了满足的性质恰好是$\Ftwo^3$的一个子空间的非零向量。对于带状图delta-拟阵，$Z_3$-二部性等价于中间图的二部性，因此该构造恢复了嵌入图的Fano平面框架。
英文摘要
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.
