Threshold for weak saturation stability

TL;DR

This paper investigates the stability threshold of the weak $K_s$-saturation number in Erdős–Rényi random graphs, establishing bounds and generalizations of previous stability results.

Contribution

It introduces a threshold for the stability of the weak $K_s$-saturation number in random graphs and provides bounds, extending prior deterministic results.

Findings

Existence of a stability threshold for weak $K_s$-saturation in $ ext{G}(n,p)$

Upper and lower bounds on the stability threshold

A general upper bound for the weak $K_s$-saturation number

Abstract

We study the weak $K_s$-saturation number of the Erd\H{o}s--R\'{e}nyi random graph $\mathbbmsl{G}(n, p)$, denoted by $\mathrm{wsat}(\mathbbmsl{G}(n, p), K_s)$, where $K_s$ is the complete graph on $s$ vertices. Kor\'{a}ndi and Sudakov in 2017 proved that the weak $K_s$-saturation number of $K_n$ is stable, in the sense that it remains the same after removing edges with constant probability. In this paper, we prove that there exists a threshold for this stability property and give upper and lower bounds on the threshold. This generalizes the result of Kor\'{a}ndi and Sudakov. A general upper bound for $\mathrm{wsat}(\mathbbmsl{G}(n, p), K_s)$ is also provided.