Sharp threshold for $K_4$-percolation

TL;DR

The paper determines the sharp threshold for $K_4$-percolation in the Erdős–Rényi graph ${\mathcal{G}}_{n,p}$, showing $p_c(n,K_4) \sim 1/\sqrt{3n\log n}$. It combines a clique-process framework with a core-decomposition strategy to isolate percolation pathways into seeds or $3$-cores and leverages tail bounds from AK17a to bound encounter probabilities. The upper bound links $K_4$-percolation to classical $2$-neighbor bootstrap percolation, while the lower bound rules out alternative percolation routes by counting irreducible percolating subgraphs and their cores. The methods yield a precise threshold, refine previous constant-factor bounds, and offer a blueprint for tackling $K_r$-percolation and related bootstrap processes in random graphs.

Abstract

We locate the critical threshold $p_c$ at which it becomes likely that the complete graph $K_n$ can be obtained from the Erdős-Rényi graph ${\cal G}_{n,p}$ by iteratively completing copies of $K_4$ minus an edge. This refines work of Balogh, Bollobás and Morris that bounds the threshold up to multiplicative constants.

Figures (1)

• Figure 1: The smallest irreducible percolating $3$-core.

Theorems & Definitions (42)

• Theorem 1
• Lemma 2: AK17a
• Theorem 3
• Definition 4
• Lemma 5
• Definition 6
• Lemma 7
• Lemma 8
• Definition 9
• Lemma 10