Introduction: on graph drawing

An old problem of geometry consists of representing a simple plane graph G by means of a collection of disks in one-to-one correspondence with the vertices of G. These disks may only intersect pairwise in at most one point, the corresponding contacts representing the edges of G. The case of disks with no prescribed shape is solved by merely drawing for each vertex v a closed curve around v and cutting the edges half way. The difficulty arises when the disks have to be of a specified shape. The famous case of circular disks, solved by the Andreev–Thurston circle packing theorem [1], involves questions of numerical analysis: the coordinates of the centers and radii are not rational, and are computed by means of convergent series. This problem is still up to date, and considered in many research works. In the present paper we will consider triangular disks.

During the last two decades a vast number of articles have been published in which probabilistic methods have been used successfully to solve combinatorial problems, and especially to obtain various asymptotic results concerning some or other characteristic of combinatorial objects. This book is aimed at readers interested in problems of this kind both from the theoretical point of view and from the point of view of possible applications.

The book may be used by students and postgraduates in combinatorics and other fields where asymptotic methods of probability theory are applied. In particular, the material contained in this book could be taught in courses on combinatorial structures such as graphs, trees and mappings with an emphasis on the asymptotic properties of their characterictics. We believe that the asymptotic results presented here provide specialists in probability theory with new examples of applications of general limit theorems.

This text assumes a standard graduate-level knowledge of probability theory and an aquaintance with typical facts drawn from a general introduction to functions of complex variables. For the reader's convenience, the relevant results of probability theory are briefly reviewed in Chapter 1. The preliminaries from combinatorial analysis are given in the introductions to each of the subsequent chapters. Readers who are interested in obtaining more detailed knowledge of the corresponding aspects of combinatorics are advised to study my book ‘Combinatorial Methods in Discrete Mathematics’ or any other basic course devoted to this subject.

The English translation differs slightly from the Russian edition of the book in the following. I have rewritten the Introduction, Subsections 2.2, 2.3 and 6.2.2. A number of minor changes have been made throughout the text to eliminate misprints and awkward proofs. Also, the list of references has been extended by the inclusion of articles and monographs devoted to relevant problems of probabilistic combinatorics which have appeared subsequent to publication of the Russian version of the book.

I am greatly indebted to Professor B. Bollobás and Professor V. Vatutin for their help and valuable advice during the preparation of the manuscript.

Introduction

This chapter is devoted to random graphs. A number of ways of introducing randomness for various classes of graph exists, one of which is to specify on some classes of graph (trees, forests, graphs of one-to-one mappings and so on) certain, as a rule uniform, probability distributions. The second way of constructing random graphs is defined by a stochastic process which gives a rule for joining a number of initially isolated vertices by edges. The third way, which is closely related to the second, is described by a random procedure of deletion of edges from a complete graph. Other methods for constructing random graphs exist but they are of little use.

Before proceeding to describe results in the field, we list a number of statements concerning the combinatorial properties of graphs that will be required in the sequel. In this chapter we deal mainly with labeled graphs and for this reason the results cited below are related, as a rule, to such combinatorial structures.

Trees

Labeled trees are, in a sense, the simplest labeled graphs. A tree is a connected graph with no cycles. A rooted tree is a tree which has a distinguished vertex called the root.

Many branches of mathematics owe a debt to classical combinatorics. This is especially true of probability theory. Plenty of good examples show how combinatorial considerations lead to very deep and difficult probabilistic results. The links between combinatorial and probabilistic problems have played an important role in forming probability theory as a mathematical discipline, and now manifest themselves in elementary courses devoted to this subject. The initial stage of the development of probability theory was characterized by the essential contribution of combinatorial methods in forming the mathematical background of the science. The current situation is quite different: well-developed probabilistic methods find a wide range of applications in solving various combinatorial problems. This is revealed in the search for asymptotic results in combinatorial analysis, where the probabilistic formulations of combinatorial problems provide the possibility to use the working system of notions of probability theory effectively and to take advantage of the powerful techniques of limit theorems in finding asymptotic formulae. It is appropriate to mention here that asymptotic results play an essential role in combinatorial analysis: they simplify calculations in problems oriented to applications and present the whole picture of investigated phenomena in a more transparent form.

For convenience of references some basic notions and facts of probability theory are listed in the first chapter of the book. Although these facts are presented in a systematic and unified form, this part of the monograph is not assumed to be a sub-stitute for a textbook on probability theory, but is directed to those readers who have some basic knowledge of the subject.

Introduction

The aim of the first two sections of this chapter is to provide a survey of the basic notions and results of probability theory which can be found in many textbooks. The concepts and theorems mentioned in Sections 1 and 2 are of an auxiliary nature and are included more for reference than for primary study. For this reason the majority of statements are given without proofs, the single exception being the theorem by Curtiss [21], which will be used frequently throughout the book.

Section 3 deals with typical examples of applications of various limit theorems to the analysis of asymptotic distributions in combinatorial problems. In terms of the properties of double generating functions we formulate rather general conditions providing asymptotic normality of certain classes of probability distributions which are met in combinatorial probability. Section 4 contains a comprehensive description of limiting distributions of random variables specified by double generating functions of the form exp{xg(t)}, where g(t) is a polynomial. In subsequent chapters, the method used to obtain the description will be extended to double generating functions of the form exp{g(x, t)}, where the function g(x, t) is not necessarily a polynomial in t.