On Emergence of Dominating Cliques in Random Graphs

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 $n$.