Geometric bijections between spanning subgraphs and orientations of a graph

Abstract

Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of $G$ and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^*$. Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called \emph{geometric bijections}. In this paper, we extend the geometric bijections to subgraph-orientation correspondences. Moreover, we extend the geometric constructions accordingly. Our proofs are purely combinatorial, even for the geometric constructions. We also provide geometric proofs for partial results, which make use of zonotopal tiling, relate to Backman, Baker, and Yuen's method, and motivate our combinatorial constructions. Finally, we explain that the main results hold for \emph{regular matroids}.