On the Domination Number of a Random Graph

Abstract

In this paper, we show that the domination number $D$ of a random graph enjoys as sharp a concentration as does its chromatic number $\chi$. We first prove this fact for the sequence of graphs $\{G(n,p_n\},\; n\to\infty$, where a two point concentration is obtained with high probability for $p_n=p$ (fixed) or for a sequence $p_n$ that approaches zero sufficiently slowly. We then consider the infinite graph $G({\bf Z}^+, p)$, where $p$ is fixed, and prove a three point concentration for the domination number with probability one. The main results are proved using the second moment method together with the Borel Cantelli lemma.