Weakly saturated random graphs

TL;DR

This paper establishes asymptotic bounds for the threshold probability at which Erdős–Rényi graphs become weakly H-saturated, advancing understanding of the phase transition for this property across various graphs.

Contribution

It provides the first asymptotic lower bounds and improved upper bounds for the weak saturation threshold in Erdős–Rényi graphs for all graphs H, especially strictly balanced ones.

Findings

Derived asymptotic lower bounds for the weak saturation threshold.

Proved upper bounds for strictly balanced graphs, including an improvement for complete graphs.

Conjectured the sharpness of the bounds up to constants.

Abstract

As introduced by Bollob\'as, a graph $G$ is weakly $H$-saturated if the complete graph $K_n$ is obtained by iteratively completing copies of $H$ minus an edge. For all graphs $H$, we obtain an asymptotic lower bound for the critical threshold $p_c$, at which point the Erd\H{o}s--R\'enyi graph ${\mathcal G}_{n,p}$ is likely to be weakly $H$-saturated. We also prove an upper bound for $p_c$, for all $H$ which are, in a sense, strictly balanced. In particular, we improve the upper bound by Balogh, Bollob{\'a}s and Morris for $H=K_r$, and we conjecture that this is sharp up to constants.