Random graphs

Abstract

Abstract In this chapter, we indicate how the general results of Chapter 2 can be used in problems involving random graphs. In Section 5.1 we consider the number of copies of a small graph G contained in Kn,p. Three cases are distinguished, counting all copies of G, induced copies of G and isolated copies of G, illustrating different ways of applying the basic Stein–Chen method. Counts of the numbers of vertices of specified degrees are considered in Section 5.2. It proves advantageous at times, when counting the vertices of degree m form large, to introduce couplings which are rather more complicated than those so far employed, necessitating the use of extensions of Stein’s method to multivariate distributional approximation along the lines described in Chapter 10. Then we study some graph colouring problems, including a general birthday problem, and questions arising therefrom concerning different versions of dissociation. Finally, we investigate two models where independent random variables are associated with the edges of a large complete graph, and random directed graphs are constructed as functions of these random variables. Related results and references on random graphs can be found in Barbour (1982), Bollobas (1985) and Janson (1986, 1987).