Two-Point Concentration of the Domination Number of Random Graphs

TL;DR

This paper proves that the domination number of random graphs is concentrated on two values for certain edge probabilities, refuting previous conjectures and introducing a novel Poisson approximation technique for bipartite graphs.

Contribution

It establishes new concentration results for the domination number in G_{n,p} and develops a novel Poisson approximation method for bipartite graphs with many isolated vertices.

Findings

Domination number concentrates on two values for p ≥ n^{-2/3+ε}.

Refutes previous conjecture about concentration for p ≤ n^{-2/3}.

Introduces a new Poisson approximation technique for bipartite graphs.

Abstract

We show that the domination number of the binomial random graph G_{n,p} with edge-probability p is concentrated on two values for p \ge n^{-2/3+\eps}, and not concentrated on two values for general p \le n^{-2/3}. This refutes a conjecture of Glebov, Liebenau and Szabo, who showed two-point concentration for p \ge n^{-1/2+\eps}, and conjectured that two-point concentration fails for p \ll n^{-1/2}. The proof of our main result requires a Poisson type approximation for the probability that a random bipartite graph has no isolated vertices, in a regime where standard tools are unavailable (as the expected number of isolated vertices is relatively large). We achieve this approximation by adapting the proof of Janson's inequality to this situation, and this adaptation may be of broader interest.