Graph Orientations and Linear Extensions

TL;DR

This paper investigates how to choose acyclic orientations of a graph to maximize the number of linear extensions of the induced partial order, providing solutions for specific graph classes and bounds for general graphs.

Contribution

It offers new results on maximizing linear extensions in acyclic orientations, including solutions for comparability graphs and odd cycles, and establishes bounds and concentration results for random graphs.

Findings

Maximizers identified for comparability graphs and odd cycles.

New bounds for the volume of the stable polytope.

Strong concentration results for graph entropy and related statistics.

Abstract

Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph and, therefore, we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem will be solved essentially for comparability graphs and odd cycles, presenting several proofs. The corresponding enumeration problem for arbitrary simple graphs will be studied, including the case of random graphs; this will culminate in 1) new bounds for the volume of the stable polytope and 2) strong concentration results for our main statistic and for the graph entropy, which hold true $a.s.$ for random graphs. We will then argue that our problem springs up naturally in the theory of graphical arrangements and graphical zonotopes.