Random Graphs, Geometry and Asymptotic Structure

Abstract

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.

Long Cycles in Locally Expanding Graphs, with Applications

“…One should notice that finding long paths, rather than cycles, in locally expanding graphs is a much easier task, see, e.g., Chapter 1 of [33] for a survey of some techniques and results available. We will encounter substantial differences between paths and cycles in this context later in the paper.…”

Perfect matchings and Hamiltonian cycles in the preferential attachment model

“…where the first inequality holds by (25) It follows that �αn� k=�εn� b k = o(1), and so a.a.s. all sets of type 2 of size at most αn satisfy the desired property, and the proof is finished for G n m−old .…”

Asymptotic Properties of Random Unlabelled Block-Weighted Graphs

“…Compare with a result for the number of connected components for the case of random unlabelled outerplanar graphs in [9, Thm. 5.1], and under a general smoothness condition given in [24,Chap. 4,Sec.…”

“…Decomposition (24) allows us to apply the substitution rule for Boltzmann samplers given in [10,Fig. 13] in order to devise a sampling procedure for graphs from the class (C b ) ω .…”