We develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of Gioan's cycle-cocycle reversal system for partial orientations. An oriented version of Dhar's burning algorithm is introduced and employed in the study of acyclicity for partial orientations. We then show that the Baker-Norine rank of a partially orientable divisor is one less than the minimum number of directed paths which need to be reversed in the generalized cycle-cocycle reversal system to produce an acyclic partial orientation. These results are applied in providing new proofs of the Riemann-Roch theorem for graphs as well as Luo's topological characterization of rank-determining sets. We prove that the max-flow mincut theorem is equivalent to the Euler characteristic description of orientable divisors and extend this characterization to the setting of partial orientations. Furthermore, we demonstrate that P ic g−1 (G) is canonically isomorphic as a P ic 0 (G)-torsor to the equivalence classes of full orientations in the cycle-cocycle reversal system acted on by directed path reversals. Efficient algorithms for computing break divisors and constructing partial orientations are presented. Contents

Let M be a regular matroid. The Jacobian group Jac(M ) of M is a finite abelian group whose cardinality is equal to the number of bases of M . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) Jac(G) of a graph G (in which case bases of the corresponding regular matroid are spanning trees of G).There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph Jac(G) and spanning trees. However, most of the known bijections use vertices of G in some essential way and are inherently "non-matroidal". In this paper, we construct a family of explicit and easy-todescribe bijections between the Jacobian group of a regular matroid M and bases of M , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of M admits a canonical simply transitive action on the set G(M ) of circuit-cocircuit reversal classes of M , and then define a family of combinatorial bijections β σ,σ * between G(M ) and bases of M . (Here σ (resp. σ * ) is an acyclic signature of the set of circuits (resp. cocircuits) of M .) We then give a geometric interpretation of each such map β = β σ,σ * in terms of zonotopal subdivisions which is used to verify that β is indeed a bijection.Finally, we give a combinatorial interpretation of lattice points in the zonotope Z; by passing to dilations we obtain a new derivation of Stanley's formula linking the Ehrhart polynomial of Z to the Tutte polynomial of M . arXiv:1701.01051v3 [math.CO] 23 Apr 20191 If e and e * are dual edges of G and G * , respectively, then given an orientation for e * we orient e by rotating the orientation of e * clockwise locally near the crossing of e and e * .

We investigate Riemann-Roch theory for directed graphs. The Riemann-Roch criteria of Amini and Manjunath is generalized to all integer lattices orthogonal to some positive vector. Using generalized notions of a v0-reduced divisor and Dhar's algorithm we investigate two chip-firing games coming from the rows and columns of the Laplacian of a strongly connected directed graph. We discuss how the "column" chip-firing game is related to directed G-parking functions and the "row" chip-firing game is related to the sandpile model. We conclude with a discussion of arithmetical graphs, which after a simple transformation may be viewed as a special class of directed graphs which will always have the Riemann-Roch property for the column chip-firing game. Examples of arithmetical graphs are provided which demonstrate that either, both, or neither of the two Riemann-Roch conditions may be satisfied for the row chip-firing game.

Gessel and Sagan [9] investigated the Tutte polynomial, T (x, y) using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is 2 g T (3, 1/2). We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is 2 n−1 T (1/2, 3). We then prove that the number of partial orientations modulo cycle reversals is 2 g T (3, 1) and the number of partial orientations modulo cut reversals is 2 n−1 T (1, 3). To prove these results, we introduce cut and cycle minimal partial orientations which provide distinguished representatives for partial orientations modulo cut and cycle reversals. These extend classes of total orientations introduced by Gioan [10], and Greene and Zaslavksy [12], and we highlight a close connection with graphic and cographic Lawrence ideals. We conclude with edge chromatic generalizations of the quantities presented, which allow for a new interpretation of the reliability polynomial for all probabilities, p with 0 < p < 1/2.

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