Graph Orientations and Linear Extensions
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 for random graphs. We will then argue that our problem springs up naturally in the theory of graphical arrangements and graphical zonotopes.
Keywords
Cite
@article{arxiv.1405.4880,
title = {Graph Orientations and Linear Extensions},
author = {Benjamin Iriarte Giraldo},
journal= {arXiv preprint arXiv:1405.4880},
year = {2015}
}
Comments
Following an editorial request, this article has been divided into two parts. This is the first part of the article originally available in arxiv:1405.4880v1, and it corresponds to Sections 1-5 of that manuscript. Several clarification comments and improvements to the original exposition were added, and typos were corrected. No new mathematical content was added. Submitted for publication