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Two-Point Concentration of the Domination Number of Random Graphs

Tom Bohman, Lutz Warnke, Emily Zhu

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.

Two-Point Concentration of the Domination Number of Random 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.
Paper Structure (12 sections, 10 theorems, 74 equations)

This paper contains 12 sections, 10 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Poisson approximation difficulty below $p=n^{-1/2}$
  4. Key ingredient: Poisson approximation beyond Janson
  5. Two-point concentration: Proof of Theorem \ref{['thm:main']}
  6. Deferred proofs and estimates
  7. Poisson approximation: Proof of Lemma \ref{['lem:Poisson']}
  8. Asymptotic independence: Proof of Lemma \ref{['lem:v1bound']}
  9. Counting estimate: Proof of Lemma \ref{['lem:v2bound']}
  10. No two-point concentration: Proof of Lemma \ref{['lem:anti']}
  11. Conclusion and open questions
  12. Appendix: Proof of the coupling result Lemma \ref{['lem:coupling']}

Key Result

Theorem 1

If $p = p(n)$ satisfies ${(\log n)^3n^{-2/3} \leq p \leq 1}$, then the domination number $\gamma(G_{n,p})$ of the binomial random graph $G_{n,p}$ is concentrated on at most two values, i.e., there exists $\hat{r} = \hat{r}(n,p)$ such that $\mathbb{P}\bigl(\gamma(G_{n,p}) \in \{\hat{r},\hat{r}+1\}\bi

Theorems & Definitions (13)

  • Theorem 1: Two-point concentration for $p \ge n^{-2/3+\varepsilon}$
  • Lemma 1: No two-point concentration for $p \le n^{-2/3}$
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm:Poisson']}
  • Lemma 2: Properties of $\hat{r}$ from glebov2015
  • Lemma 3: Main technical result
  • Lemma 4: Asymptotic Independence
  • Lemma 5: Counting Estimate
  • Lemma 6
  • Lemma 7: glebov2015
  • ...and 3 more