When does a tree activate the random graph?

TL;DR

This paper introduces a novel topological approach to analyze weak saturation in graphs, providing tight bounds and thresholds for the appearance of saturated trees and connectivity properties in random graphs.

Contribution

It uncovers a new topological connection to weak saturation, enabling the derivation of tight bounds and thresholds where algebraic methods are less effective.

Findings

Critical probability for $K_3$-saturating trees is of order $n^{-1/3-o(1)}$

Determines the threshold for trees with bounded diameter up to a constant factor

Improves bounds on the connectivity threshold of clique complexes in random graphs

Abstract

Let $F$ and $G$ be two graphs. A spanning subgraph $H$ of $G$ is called weakly $F$-saturated if one can add to $H$ the edges of $G \setminus H$ in some order, so that whenever a new edge is added, a new copy of $F$ is formed. Obtaining lower bounds for the minimum size $\mathrm{wsat}(G,F)$ of such an $H$ is a classical problem in extremal combinatorics. In particular, in the past 40 years, various algebraic tools have been developed to prove lower bounds on the weak saturation number $\mathrm{wsat}(G,F)$. Our paper uncovers a new connection of weak saturation to topology of clique complexes, that allows to prove tight lower bounds in some cases when the algebraic tools are not efficient. It is easy to see that the smallest $K_3$-saturating graphs in $K_n$ are trees, thus $\mathrm{wsat}(K_n,K_3)=n-1$. In 2017, Kor\'andi and Sudakov proved that this is also the case in dense random graphs $G\sim G_{n,p}$, $p=\mathrm{const}\in(0,1)$, and posed the question of determining the smallest $p$ for which $G_{n,p}$ contains a $K_3$-saturating tree with high probability. Using the new topological connection, we show that this critical $p$ is of order $n^{-1/3-o(1)}$. Inspired by Gromov's local-to-global principle for hyperbolic groups, we further develop our topological approach and determine the critical probability up to a constant factor, for trees with diameter at most $n^{c}$, for some $c>0$. The new connection also enables us to improve the best known upper bound on the threshold probability for simple connectivity of the 2-dimensional clique complex of $G_{n,p}$, due to Kahle.