Abstract
Let G be a connected finite graph. The Picard group of G, the set of spanning trees of G, and the equivalence classes of orientations of G up to cycle-cocycle reversal are equinumerous. We study some interesting combinatorial and geometric correspondences among them. In the first part of this dissertation, we study some torsor structures on the set of spanning trees for the Picard group. A torsor for a group P is a set S together with a simply transitive action of P on S. Such a structure can exist only if P and S are equinumerous. There are several known torsor structures on the set of the spanning trees of G for the Picard group of G. The rotor-routing torsor and the Bernardi torsor are two such torsors constructed combinatorially. Both of the constructions require to endow G with a ribbon structure. Matthew Baker and Yao Wang proved that the rotor-routing torsor and the Bernardi torsor coincide when G is planar. We prove the conjecture raised by them that the two torsors disagree when G is non-planar.
In the second part of this dissertation, we study some correspondences between spanning trees and equivalence classes of orientations. Two orientations of G are said to be in the same cycle-cocycle reversal (equivalence) class if one can obtain one orientation from the other by reversing some directed cycles and directed cocycles. The set of the cycle-cocycle reversal classes of G is known to be a canonical torsor for the Picard group. Spencer Back- man, Matthew Baker, and Chi Ho Yuen have constructed a family of bijections between the spanning trees of G and the cycle-cocycle reversal classes, which induces another tor- sor structure on the set of spanning trees. Their proof makes use of zonotopal subdivisions and the bijections are called geometric bijections. We extend the geometric bijections to subgraph-orientation correspondences. Our extended bijections specialize to bijections be- tween spanning forests and cycle reversal classes of orientations, and bijections between spanning connected subgraphs and cocycle reversal classes of orientations.