Random acyclic orientations of graphs

II Preface This is a Master's thesis written at the department of Mathematics at the Royal Institute of Technology (KTH) in Stockholm, supervised by Svante Linusson. It is assumed that the reader has studied mathematics at an undergraduate level, and that he is well acquainted with some basic graph theory concepts such as paths and cycles. III Summary This paper is about acyclic orientations of graphs and various models for randomly oriented graphs. Acyclic orientations are important objects within Graph Theory. We will briefly go through some of the most important interpretations and applications, but focus on the pure mathematical results. Richard Stanley has shown that much information about acyclic orientations is encoded in the chromatic polynomial. For instance, the total number of acyclic orientations of a given graph is equal to , [11], and the number of acyclic orientations with a unique source is equal to 'the linear coefficient of '. Randomly oriented graphs have received relatively little attention compared to its undirected counterparts, such as (a generalization of the familiar), where each edge of a graph is removed with probability. In this paper we will define and analyze various probability spaces (models) over directed graphs. The main problem will be to study " The probability that two given vertices and are connected by some directed path ". Secondary problems include analyzing the probability for having a unique source or sink, as well as analyzing how the existence of certain paths correlate with the existence of other paths. We shall mainly be working with 3 different models. One of which, perhaps the most natural one, is uniform and based on [1] by Sven Erick Alm, Svante Janson and Svante Linusson. The second one is based on [3] by Athanasiadis Christos A and Diaconis Persi, where a random walk on a certain hyperplane arrangement produces a distribution of acyclic orientations. The third one, denoted is completely new; it resembles , but consists of directed acyclic graphs (DAGs). In we will provide an exact recursive formula for , and then give lower and upper bounds for the percolation threshold which is conjectured to be √. When is held fixed and , we will show that in the probability of having a unique source tends to some constant , with for .