Rational values of the weak saturation limit

TL;DR

This paper characterizes all rational values of the weak saturation limit constant for graphs, showing that it can be any rational number at least 1.5, thus advancing understanding of graph saturation properties.

Contribution

It provides a complete characterization of the rational values of the weak saturation limit constant for all graphs, including the proof that any rational number ≥ 1.5 can be realized.

Findings

The weak saturation limit constant $w_F$ can be any rational number at least 1.5.

The paper characterizes all possible rational values of $w_F$ for graphs.

It extends the understanding of weak saturation in graph theory.

Abstract

Given a graph $F$, a graph $G$ is weakly $F$-saturated if all non-edges of $G$ can be added in some order so that each new edge introduces a copy of $F$. The weak saturation number $\operatorname{wsat}(n, F)$ is the minimum number of edges in a weakly $F$-saturated graph on $n$ vertices. Bollob\'as initiated the study of weak saturation in 1968 to study percolation processes, which originated in biology and have applications in physics and computer science. It was shown by Alon that for each $F$, there is a constant $w_F$ such that $\operatorname{wsat}(n, F) = w_Fn + o(n)$. We characterize all possible rational values of $w_F$, proving in particular that $w_F$ can equal any rational number at least $\frac 32$.