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.

It is regrettable that a book, once published and on the way to starting a life of its own, can no longer bear witness to the painful choices that the author had to face in the course of his writing. There are choices that confront the writer of every book: who is the intended audience? who is to be proved wrong? who will be the most likely critic? Most of us have indulged in the idle practice of drafting tables of contents of books we know will never see the light of day. In some countries, some such particularly imaginative drafts have actually been sent to press (though they may not be included among the author's list of publications).

In mathematics, however, the burden of choice faced by the writer is so heavy as to turn off all but the most courageous. And of all mathematics, combinatorics is nowadays perhaps the hardest to write on, despite an eager audience that cuts across the party lines. Shall an isolated special result be granted a section of its own? Shall a fledgling new theory with as yet sparse applications be gingerly thrust in the middle of a chapter? Shall the author yield to one of the contrary temptations of recreational math at one end, and categorical rigor at the other? or to the highly rewarding lure of the algorithm?

Inclusion–Exclusion

Roughly speaking, a “sieve method” in enumerative combinatorics is a method for determining the cardinality of a set S that begins with a larger set and somehow subtracts off or cancels out the unwanted elements. Sieve methods have two basic variations: (1) We can first approximate our answer by an overcount, then subtract off an overcounted approximation to our original error, and so on, until after finitely many steps we have “converged” to the correct answer. This is the combinatorial essence of the Principle of Inclusion–Exclusion, to which this section and the next four are devoted. (2) The elements of the larger set can be weighted in a natural combinatorial way so that the unwanted elements cancel out, leaving only the original set S. We discuss this technique in Sections 2.5–2.7.

The Principle of Inclusion–Exclusion is one of the fundamental tools of enumerative combinatorics. Abstractly, the Principle of Inclusion–Exclusion amounts to nothing more than computing the inverse of a certain matrix. As such it is simply a minor result in linear algebra. The beauty of this principle lies not in the result itself, but rather in its wide applicability. We will give several examples of problems that can be solved by the Principle of Inclusion–Exclusion, some in a rather subtle way. First we state the principle in its purest form.

Enumerative combinatorics is concerned with counting the number of elements of a finite set S. This definition, as it stands, tells us little about the subject since virtually any mathematical problem can be cast in these terms. In a genuine enumerative problem, the elements of S will usually have a rather simple combinatorial definition and very little additional structure. It will be clear that S has many elements, and the main issue will be to count (or estimate) them all and not, for example, to find a particular element. Of course there are many variants of this basic problem that also belong to enumerative combinatorics and that will appear throughout this book.

There has been an explosive growth in combinatorics in recent years, including enumerative combinatorics. One important reason for this growth has been the fundamental role that combinatorics plays as a tool in computer science and related areas. A further reason has been the prodigious effort, inaugurated by G.-C. Rota around 1964, to bring coherence and unity to the discipline of combinatorics, particularly enumeration, and to incorporate it into the mainstream of contemporary mathematics. Enumerative combinatorics has been greatly elucidated by this effort, as has its role in such areas of mathematics as finite group theory, representation theory, commutative algebra, algebraic geometry, and algebraic topology.

This book has three intended audiences and serves three different purposes. First, it may be used as a graduate-level introduction to a fascinating area of mathematics.