Random Graphs, Geometry and Asymptotic Structure

The theory of random graphs is a vital part of the education of any researcher entering the fascinating world of combinatorics. However, due to their diverse nature, the geometric and structural aspects of the theory often remain an obscure part of the formative study of young combinatorialists and probabilists. Moreover, the theory itself, even in its most basic forms, is often considered too advanced to be part of undergraduate curricula, and those who are interested usually learn it mostly through self-study, covering a lot of its fundamentals but little of the more recent developments. This 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 and introduces the reader to the diversity and depth of the methods that have been devised in this context.

Editors' introduction; Part I. Long Paths and Hamiltonicity in Random Graphs: 1. Introduction; 2. Tools; 3. Long paths in random graphs; 4. The appearance of Hamilton cycles in random graphs; References for Part I; Part II. Random Graphs from Restricted Classes: 1. Introduction; 2. Random trees; 3. Random graphs from block-stable classes; References for Part II; Part III. Lectures on Random Geometric Graphs: 1. Introduction; 2. Edge counts; 3. Edge counts: normal approximation; 4. The maximum degree; 5. A sufficient condition for connectivity; 6. Connectivity and Hamiltonicity; 7. Solutions to exercises; References for Part III; Part IV. On Random Graphs from a Minor-closed Class: 1. Introduction; 2. Properties of graph classes; 3. Bridge-addability, being connected and the fragment; 4 Growth constants; 5. Unlabelled graphs; 6. Smoothness; 7. Concluding remarks; References for Part IV; Index.