Weakly saturated random graphs
Zsolt Bartha, Brett Kolesnik
TL;DR
This work analyzes the threshold for weakly $H$-saturated (or $H$-percolating) behavior in Erdős–Rényi graphs. It delivers a universal lower bound on the critical threshold $p_c(n,H)$ that applies to all graphs $H$ by exploiting witness-graph structures and the $\lambda_*$ parameter, and proves a sharper upper bound for strictly balanced $H$ via induced $H$-ladders and a two-round second moment argument, followed by sprinkling. Consequently, for balanced graphs $H$ one has $p_c(n,H)=n^{-1/\lambda+o(1)}$, and the upper bound is improved for $H=K_r$ with $r\ge5$, advancing beyond the prior results of BBM12. The methods combine a general WSA/REA framework to bound lower thresholds with a ladder-based, second-moment approach to upper thresholds, clarifying how the graph structure (via $\lambda$ and $\lambda_*$) governs percolation behavior. The results also establish a near-complete dichotomy between balanced and strictly balanced graphs in terms of threshold scaling, and pose conjectures about sharpness up to constants in the strictly balanced regime.
Abstract
As introduced by Bollobás, 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 Erdős--Rényi 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{á}s and Morris for $H=K_r$, and we conjecture that this is sharp up to constants.