Sieve Methods in Random Graph Theory

Abstract

In this paper, we apply the Turan sieve and the simple sieve developed by R. Murty and the first author to study problems in random graph theory. In particular, we obtain upper and lower bounds on the probability of a graph on n vertices having diameter 2 (or diameter 3 in the case of bipartite graphs) with edge probability p(n) where the edges are chosen independently . An interesting feature revealed in these results is that the Turan sieve and the simple sieve `almost completely' complement each other. As a corollary to our result, we note that the probability of a random graph having diameter 2 approaches 1 as n approaches infinity for constant edge probability p(n)=1/2. This is an appendix of a shorter version of this paper.