INTRODUCTION

The subject of graph decompositions is a vast and sprawling topic, one which we certainly cannot begin to cover in a paper of this length. Indeed, recently a number of survey articles and several books have appeared, each devoted to a particular subtopic within this domain (e.g., see [Fi-Wi], [Gr-Rot-Sp],[So 1],[Do-Ro]).

What we will attempt to do in this report is twofold. First, we will try to give a brief overall view of the landscape, mentioning various points of interest (to us) along the way. When possible, we will provide the reader with references in which much more detailed discussions can be found. Second, we will focus more closely on a few specific topics and results, usually for which significant progress has been made within the past few years. We will also list throughout various problems, questions and conjectures which we feel are interesting and/or contribute to a clearer understanding of some of the current obstacles remaining in the subject.

Notation

By a graph G we will mean a (finite) set V = V(G), called the vertices of G together with a set E = E(G) of (unordered) pairs of vertices of G, called the edges of G.

Let H denote a family of graphs. By an H-decomposition of G we mean a partition of E(G) into disjoint sets E(H.) such that each of the graphs Hi induced by the edge set E(Hi) is isomorphic to a graph in H.

The aim of this review is to highlight some of the fundamental results about random graphs, mostly in areas I am particularly interested in. Though a fair number of references are given, the review is far from complete even in the topics it covers. Furthermore, very few of the proofs are indicated. The exception is the last section, which concerns random regular graphs. This section contains some very recent results and we present some proofs in a slightly simplified form.

The study of random graphs was started by Erdòs [33], who applied random graph techniques to show the existence of a graph of large chromatic number and large girth. A little later Erdös and Rényi [38] investigated random graphs for their own sake. They viewed a graph as an organism that develops by acquiring more and more edges in a random fashion. The question is at what stage of development a graph is likely to have a given property. The main discovery of Erdös and Rényi was that many properties appear rather suddenly. In the last twenty years many papers have been written about random graphs; some of them, in the vein of [33], tackle traditional graph problems by the use of random graphs, and others, in fact the majority, study the standard invariants of random graphs in the vein of [38]. Of course the two trends cannot really be separated for deep applications are impossible without detailed knowledge of random graphs.

Abstract

The enumeration of graphs and other structures satisfying a duality condition, such as self-complementarity, is surveyedo A modification of Burnside's lemma due to de Bruijn is presented in order to unify and simplify the treatment of such problems. Some new equalities between classes of graphs and digraphs are found which seem not to be explained by natural 1-1 correspondences. Also some new natural 1–1 correspondences are derived using the modified Burnside lemma. Asymptotic analyses of the exact numbers are reviewed, and some recent results described.

Introduction

Methods for counting graphs and related structures which satisfy a duality condition are well-established in the literature. A duality condition is defined by invariance up to isomorphism under some operation. Complementation is an operation which has often been considered. Self-complementary structures enumerated include graphs and digraphs [Re63], tournaments [Sr70], n-plexes [Pa73a], m-ary relations [Wi74], multigraphs [Wi78], eulerian graphs [Ro69], bipartite graphs [Qu79], sets [B59 and B64], and boolean functions [Ni59, El60, Ha63, Ha64, and PaR-A]. Closely related are 2-colored or signed structures invariant under color interchange or sign interchange. These take in 2-colored graphs [HP63 and Ha7 9], graphs in which points, lines, or points and lines are signed [HPRS77], signed graphs under weak isomorphism [So80], 2-colored polyhedra [R27 and KnPR75], and necklaces [PS77 and Mi78]. The converse of a digraph results when all orientations of arcs are reversed.

Introduction

Every undergraduate course in graph theory mentions basic results about finite planar graphs – Kuratowski's criterion for embeddability, Euler's Theorem, and so on. However the corresponding results for infinite graphs seem to be little known. It turns out that the concept of embeddability in the plane has many ramifications and variants in the infinite case, and one of the purposes of this exposition is to survey these. For the most part results will only be quoted and no proofs given – for these the reader is referred to the literature listed in the bibliography.

In this survey we aim to show how fruitful is the interaction between the theories of finite and of infinite planar graphs. Results from one of these fields often inspire nontrivial problems in the other, and frequently suggest analogous questions about graphs embeddable in manifolds of arbitrary genus.

Although it may seem foreign to the subject, especially to those only interested in problems of a strictly combinatorial or topological nature, quite a large part of what we shall do will be metrical in character. There are several reasons for this. For example, in our discussion of Euler's Theorem and its variants in Section 3, the results are not true unless we impose quite strong restrictions on the kinds of graph we are considering – and we only know how to formulate these restrictions in metrical terms.