Equivalences on Acyclic Orientations

TL;DR

This paper explores the structure of acyclic orientations of graphs under group actions, characterizes their equivalence classes, and connects these to graph invariants like the Tutte polynomial, with implications for combinatorics and algebra.

Contribution

It constructs graphs encoding equivalence classes of acyclic orientations, characterizes their structure, and relates these classes to the Tutte polynomial and poset structures.

Findings

Connected components encode equivalence classes of orientations.

Delta(Y) can be derived from kappa(Y).

Kappa(Y) relates to Tutte polynomial evaluation.

Abstract

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 ~kappa and ~delta, 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 kappa(Y) and delta(Y), respectively. We characterize the structure of C(Y) and D(Y), show how delta(Y) can be derived from kappa(Y), and give enumeration results for kappa(Y). Moreover, we show how to associate a poset structure to each kappa-equivalence class, and we characterize these posets. This allows us to create a bijection from Acyc(Y)/~kappa to the union of Acyc(Y')/~kappa and Acyc(Y'')/~kappa, Y' and Y'' denote edge deletion and edge contraction for a cycle-edge in Y, respectively, which in turn shows that kappa(Y) may be obtained by an evaluation of the Tutte polynomial at (1,0).