Weak saturation stability

TL;DR

This paper investigates the stability of weak saturation numbers in complete graphs, proving stability for certain pattern graphs and establishing thresholds for specific cases like stars.

Contribution

It introduces the concept of stability in weak saturation, proving it for all pattern graphs with local percolation properties and identifying thresholds for star graphs.

Findings

wsat(K_n,H) remains stable under random edge removal for certain H

Established stability for cliques and bipartite graphs

Derived threshold probabilities for weak saturation of star graphs

Abstract

The paper studies wsat$(G,H)$ which is the minimum number of edges in a weakly $H$-saturated subgraph of $G$. We prove that wsat$(K_n,H)$ is `stable' - remains the same after independent removal of every edge of $K_n$ with constant probability - for all pattern graphs $H$ such that there exists a `local' set of edges percolating in $K_n$. This is true, for example, for cliques and complete bipartite graphs. We also find a threshold probability for the weak $K_{1,t}$-saturation stability.