Geometric bijections between spanning subgraphs and orientations of a graph

Abstract

AbstractLet be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy‐to‐describe bijections between spanning trees of and ‐compatible orientations, where the ‐compatible orientations are the representatives of equivalence classes of orientations up to cycle–cocycle reversal that are determined by a cycle signature and a cocycle signature . Their bijections are geometric because the construction comes from zonotopal subdivisions. In this paper, we extend the geometric bijections to subgraph‐orientation correspondences. Moreover, we extend the geometric constructions accordingly. Our proofs are combinatorial, which do not make use of the zonotopes. 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 regular matroids.