Sieve Methods in Random Graph Theory

TL;DR

This paper applies sieve methods to analyze probabilities of certain diameters in random graphs, revealing that the probability of diameter 2 approaches 1 for constant edge probability as the number of vertices grows.

Contribution

It introduces the use of Turan and simple sieve methods to derive bounds on diameters in random graphs, showing their complementary effectiveness.

Findings

Probability of diameter 2 tends to 1 for p(n)=1/2 as n→∞

Turan and simple sieves complement each other in analysis

Bounds on diameter probabilities for bipartite and non-bipartite graphs

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.