In this second edition of our book we have tried to maintain the same structure as the first edition, namely a text which, although not providing an exhaustive coverage of graph symmetries and reconstruction, provides a detailed coverage of some particular areas (generally motivated by our own research interest), which is not a haphazard collection of results but which presents a clear pathway through this thick forest. And our aim remains that of producing a text which can relatively quickly guide the reader to the point of being able to understand and carry out research in the topics which we cover.

Among the additions in this edition we point out the use of the free computer programs GAP, GRAPE and Sage to construct and investigate some wellknown graphs, including examples with properties like being semisymmetric, a topic which was treated in the first edition but for which examples are not easy to construct ‘by hand’. We have also updated some chapters with new results, improved the presentation and proofs of others, and introduced short treatments of topics such as character theory of abelian groups and their Cayley graphs to emphasise the connection between graph theory and other areas of mathematics.

We have corrected a number of errors which we found in the first edition, and for this we would like to thank colleagues who have pointed out several of them, particularly Bill Kocay, Virgilio Pannone and Alex Scott.

A special thanks goes to Russell Mizzi for help with overhauling Chapter 6, where we also introduce the new idea of two-fold isomorphisms, and to Leonard Soicher and Matan Zif-Av for several helpful tips regarding the use of GAP and GRAPE.

The second author would like to thank the Politecnico di Milano for giving him the opportunity, by means of a sabbatical, to focus on the work needed to complete the current edition of this book. He also thanks the University of Malta for its kind hospitality during this sabbatical. The authors will maintain a list of corrections and addenda at http://staff.um.edu.mt/josef.lauri.

This book arose out of lectures given by the first author to Masters students at the University of Malta and by the second author at the Università Cattolica di Brescia.

This book is not intended to be an exhaustive coverage of graph theory. There are many excellent texts that do this, some of which are mentioned in the References. Rather, the intention is to provide the reader with a more in-depth coverage of some particular areas of graph theory. The choice of these areas has been largely governed by the research interests of the authors, and the flavour of the topics covered is predominantly algebraic, with emphasis on symmetry properties of graphs. Thus, standard topics such as the automorphism group of a graph, Frucht's Theorem, Cayley graphs and coset graphs, and orbital graphs are presented early on because they provide the background for most of the work presented in later chapters. Here, more specialised topics are tackled, such as graphical regular representations, pseudosimilarity, graph products, Hamiltonicity of Cayley graphs and special types of vertex-transitive graphs, including a brief treatment of the difficult topic of classifying vertex-transitive graphs. The last four chapters are devoted to the Reconstruction Problem, and even here greater emphasis is given to those results that are of a more algebraic character and involve the symmetry of graphs. A special chapter is devoted to graph products. Such operations are often used to provide new examples from existing ones but are seldom studied for their intrinsic value.

Throughout we have tried to present results and proofs, many of which are not usually found in textbooks but have to be looked for in journal papers. Also, we have tried, where possible, to give a treatment of some of these topics that is different from the standard published material (for example, the chapter on graph products and much of the work on reconstruction).

The theory of random graphs was established during the 1950s through the pioneering work of Gilbert and subsequently of Erdős and Rényi who set its foundations. Since then, the theory has been developed vastly and is by now a central area of combinatorics. Numerous, often unexpected, ramifications have emerged, which link it to diverse areas of mathematics such as number theory, combinatorial optimization and probability theory. Since its beginning, the study of geometric and topological aspects of random graphs has become the meeting point between combinatorics and areas of probability theory, such as percolation theory and stochastic processes. Nowadays, this interface has been consolidated through numerous deep results. This has led to applications in other scientific disciplines including telecommunications, astronomy, statistical physics, biology and computer science, as well as much more recent developments such as the study of social and biological networks.

The present book is the outcome of a short course that took place at the School of Mathematics of the University of Birmingham in August 2013 and was supported by the London Mathematical Society and the Engineering and Physical Sciences Research Council. Its aim was to provide a concise overview of recent trends in the theory of random graphs, ranging from classical structural problems to geometric and topological aspects, and to introduce the participants to new powerful complex–analytic techniques and stochastic models that have led to recent breakthroughs in the field.

The theory of random graphs is nowadays part and parcel of the education of any young researcher entering the fascinating world of combinatorics. However, due to their interdisciplinary nature, the geometric and structural aspects of the theory often remain an obscure part of the education of young researchers. Moreover, the theory itself, even in its most basic forms, is often considered quite advanced to be part of undergraduate curricula, and those interested, usually learn it mostly through self-study, covering a lot of its fundamentals but not much of the more recent developments. The present book provides a self-contained and concise introduction to recent developments and techniques for classical problems in the theory of random graphs. Moreover, it covers geometric and topological aspects of the theory of random graphs and introduces the reader to the diversity and depth of the methods that have been invented in this context.

Introduction

Long paths and Hamiltonicity are certainly among the most central and researched topics of modern graph theory. It is thus only natural to expect that they will take a place of honor in the theory of random graphs. And indeed, the typical appearance of long paths and of Hamilton cycle is one of the most thoroughly studied directions in random graphs, with a great many diverse and beautiful results obtained over the past fifty or so years.

In this survey we aim to cover some of the most basic theorems about long paths and Hamilton cycles in the classical models of random graphs, such as the binomial random graph or the random graph process. By no means should this text be viewed as a comprehensive coverage of results of this type in various models of random graphs; the reader looking for breadth should rather consult research papers or a recent monograph on random graphs by Frieze and Karoński [1]. Instead, we focus on simplicity, aiming to provide accessible proofs of several classical results on the subject and showcasing the tools successfully applied recently to derive new and fairly simple proofs, such as applications of the Depth First Search (DFS) algorithm for finding long paths in random graphs and the notion of boosters.

Although this chapter is fairly self-contained mathematically, basic familiarity and hands-on experience with random graphs would certainly be of help for the prospective reader. The standard random graph theory monographs of Bollobás [2] and of Janson et al. [3] certainly provide (much more than) the desired background.

This chapter is based on a mini-course with the same name, delivered by the author at the LMS-EPSRC Summer School on Random Graphs, Geometry, and Asymptotic Structure, organized by Dan Hefetz and Nikolaos Fountoulakis at the University of Birmingham in the summer of 2013. The author would like to thank the course organizers for inviting him to deliver the mini-course, and for encouraging him to create lecture notes for the course, which eventually served as a basis for the present chapter.