Georgia Institute of Technology College of Sciences

A fourientation of a graph is a choice for each edge of whether to orient it in either direction, bidirect it, or leave it unoriented. I will present joint work with Sam Hopkins where we describe classes of fourientations defined by properties of cuts and cycles whose cardinalities are given by generalized Tutte polynomial evaluations of the form: (k+l)^{n-1}(k+m)^g T (\frac{\alpha k + \beta l +m}{k+l}, \frac{\gamma k +l + \delta m}{k+m}) for \alpha,\gamma \in {0,1,2} and \beta, \delta \in {0,1}. We also investigate classes of 4-edge colorings defined via generalized notions of internal and external activity, and we show that their enumerations agree with those of the fourientation classes. We put forth the problem of finding a bijection between fourientations and 4-edge-colorings which respects all of the given classes. Our work unifies and extends earlier results for fourientations due to myself, Gessel and Sagan, and Hopkins and Perkinson, as well as classical results for full orientations due to Stanley, Las Vergnas, Greene and Zaslavsky, Gioan, Bernardi and others.