Fourientations and the Tutte polynomial

Abstract

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 formfor α, γ ∈ {0, 1, 2} and β, δ ∈ {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 , . arXiv:1205. 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 algebra, and the reliability polynomial.

Tutte polynomials for directed graphs

“…The case of Theorem 7.5 corresponding to unoriented graphs was first proved by Gessel and Sagan [15]. This result for undirected graphs was also rediscovered as part of the theory of fourientations for the Tutte polynomial developed in [4,5,6]. Recall that in a fourientation of a graph G, the edges of G can be oriented in either direction (1-way edges), in both direction (2-way edges), or in no direction (0-way edges).…”

“…For (h), observe that B D (q, 1, 1) counts all q-colorings, hence B D (q, 1, 1) = q |V | . Moreover deg q (B D (q, y, z)) cannot be more than |V | by (5). Clearly deg y (B D (q, y, y)) ≤ |A|, and considering proper q-colorings gives equality for loopless digraphs.…”

“…Moreover, B D (q, 0, 0) counts q-colorings which are constant on each connected components, and there are q c(D) such q-colorings. Property (j) and (k) follow directly from (5). Lastly, (l) is clear from the definitions upon identifying each 2-coloring f with the subset of vertices…”

Morphisms and Order Ideals of Toric Posets

“…We will say that this is a preposet over G, because it can be described by an (not necessarily acyclic) orientation ω π of G, where one allows bi-directed edges. These were called Type B fourientations in a recent paper by Backman and Hopkins [17], though we will simply call them "orientations" since we don't refer to them often, and when we do, it should always be clear from the context that bi-directed edges are allowed. The notation ω π reflects the fact that this orientation can be constructed by starting with some ω ∈ Acyc(G) and then making each edge bidirected if both endpoints are contained in the same block of π.…”

Zonotopal algebra and forward exchange matroids

“…It is spanned by products of linear forms and it can be written as the Macaulay inverse system (or kernel) of an ideal generated by powers of linear forms [15]. Ideals of this type and their inverse systems are also studied in the literature on fat point ideals [30,31] and graph orientations [4].…”