Graph Orientations and Linear Extensions

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 ori...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Author Benjamin Iriarte Giraldo
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 14.02.2015
Subjects
Apexes
Enumeration
Graph theory
Graphs
Set theory
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422