On Emergence of Dominating Cliques in Random Graphs

TL;DR

This paper studies the emergence of dominating cliques in Erdős-Rényi random graphs, revealing a phase transition depending on the edge probability p, with detailed analysis of the critical probability range.

Contribution

It establishes the conditions under which dominating cliques almost surely appear or not in Erdős-Rényi graphs, highlighting a phase transition phenomenon.

Findings

Dominating cliques emerge almost surely when p > 1/2.

Dominating cliques do not emerge almost surely when p ≤ (3 - √5)/2.

The critical probability range is analyzed through sub-logarithmic growth of clique size.

Abstract

Emergence of dominating cliques in Erd\"os-R\'enyi random graph model ${\bbbg(n,p)}$ is investigated in this paper. It is shown this phenomenon possesses a phase transition. Namely, we have argued that, given a constant probability $p$, an $n$-node random graph $G$ from ${\bbbg(n,p)}$ and for $r= c \log_{1/p} n$ with $1 \leq c \leq 2$, it holds: (1) if $p > 1/2$ then an $r$-node clique is dominating in $G$ almost surely and, (2) if $p \leq (3 - \sqrt{5})/2$ then an $r$-node clique is not dominating in $G$ almost surely. The remaining range of probability $p$ is discussed with more attention. A detailed study shows that this problem is answered by examination of sub-logarithmic growth of $r$ upon $n$.