Equivalences on Acyclic Orientations

The cyclic and dihedral groups can be made to act on the set Acyc(Y ) of acyclic orientations of an undirected graph Y , and this gives rise to the equivalence relations ∼κ and ∼δ, respectively. These two actions and their corresponding equivalence classes are closely related to combinatorial problems arising in the context of Coxeter groups, sequential dynamical systems, the chip-firing game, and representations of quivers. In this paper we construct the graphs C(Y ) and D(Y ) with vertex sets Acyc(Y ) and whose connected components encode the equivalence classes. The number of connected components of these graphs are denoted κ(Y ) and δ(Y ), respectively. We characterize the structure of C(Y ) and D(Y ), show how δ(Y ) can be derived from κ(Y ), and give enumeration results for κ(Y ). Moreover, we show how to associate a poset structure to each κ-equivalence class, and we characterize these posets. This allows us to create a bijection from Acyc(Y )/∼κ to Acyc(Y )/∼κ ∪ Acyc(Y )/∼κ, where Y ′ and Y ′′ denote edge deletion and edge contraction for a cycle-edge in Y , respectively, which in turn shows that κ(Y ) may be obtained by an evaluation of the Tutte polynomial at (1, 0).