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 $\tau$, let $Z_3(D,\tau )$ 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 $Z_3(D,\tau )$ 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.