Riemann-Roch Theory for Graph Orientations

Abstract: 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 min-cut 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 $Pic^{g-1}(G)$ is canonically isomorphic as a $Pic^{0}(G)$-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.