Fourientations and the Tutte Polynomial

A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we investigate properties of cuts and cycles in fourientations which give trivariate generating functions that are generalized Tutte polynomial evaluations of the form $$(k+m)^{n-1}(k+l)^gT\left(\frac{\alpha k + \beta l + m}{k+m},\frac{\gamma k + l + \delta m}{k+l}\right)$$ for $\alpha,\gamma \in \{0,1,2\}$ and $\beta, \delta \in \{0,1\}$. We introduce an intersection lattice of 64 cut-cycle fourientation classes enumerated by generalized Tutte polynomial evaluations of this form. We prove these enumerations using a single deletion-contraction argument and classify axiomatically the set of fourientation classes to which our deletion-contraction argument applies. This work unifies and extends earlier results for fourientations due to Gessel and Sagan, and results for partial orientations due to the first author, and the second author and David Perkinson, as well as results for total orientations due to many authors. We conclude by describing how these classes of fourientations relate to geometric, combinatorial, and algebraic objects including bigraphical arrangements, cycle-cocycle reversal systems, graphic Lawrence ideals, Riemann-Roch theory for graphs, zonotopal algebras, and the reliability polynomial.