The previous chapter dealt with the existence of small subgraphs of a fixed size. In this chapter, we concern ourselves with the existence of large subgraphs, most notably perfect matchings and Hamilton cycles. Having dealt with perfect matchings, we turn our attention to long paths in sparse random graphs, i.e., in those where we expect a linear number of edges. We next study one of the most celebrated and difficult problems of random graphs: the existence of a Hamilton cycle in a random graph. In the last section of this chapter, we consider the general problem of the existence of arbitrary spanning subgraphs in a random graph

In this chapter, we mainly explore how the typical component structure evolves as the number of edges m increases. The following statements should be qualified with the caveat, w.h.p. The evolution of Erdős–Rényi–Gilbert type random graphs has clearly distinguishable phases. The first phase, at the beginning of the evolution, can be described as a period when a random graph is a collection of small components which are mostly trees. Next, a random graph passes through a phase transition phase when a giant component, of order comparable with the order of random graph, starts to emerge.

Large real-world networks although being globally sparse, in terms of the number of edges, have their nodes/vertices connected by relatively short paths. In addition, such networks are locally dense, i.e., vertices lying in a small neighborhood of a given vertex are connected by many edges. This observation is called the “small-world” phenomenon, and it has generated many attempts, both theoretical and experimental, to build and study appropriate models of small-world networks. The first attempt to explain this phenomenon and to build a more realistic model was introduced by Watts and Strogatz in 1998 followed by the publication of an alternative approach by Kleinberg in 2000. The current chapter is devoted to the presentation of both models.

In this chapter, we look first at the diameter of random graphs, i.e., the extreme value of the shortest distance between a pair of vertices. Then we look at the size of the largest independent set and the related value of the chromatic number. One interesting feature of these parameters is that they are often highly concentrated.