Rational values of the weak saturation limit
Ruben Ascoli, Xiaoyu He
TL;DR
The paper resolves the long-standing problem of characterizing the rational values that the weak saturation limit $w_F$ can attain for a graph $F$. It introduces a pivotal link between $w_F$ and the isoperimetric quantity $gamma_F$, then develops a two-pronged analysis: a sparse regime where $w_F< ext{deg}_F/2$ and a dense regime covering $gamma_F$ in wide rational intervals. Sparse constructions show $w_F$ cluster at $ ext{deg}_F/2-1/k$ for suitable $k$, while dense constructions, including expander-based and deterministic subdivision schemes, realize any rational $w$ in the admissible ranges for $ ext{deg}_F= ext{2,3,4}$ and for $ ext{deg}_F o ext{6}+$ via carefully glued graphs. The results yield a near-complete rational classification of $w_F$ and provide techniques (rotation, activation/ownership, and isoperimetric control) that may extend to hypergraphs. They also address Tuza's conjecture in the sparse regime and present notable insights and open questions about the nature and attainability of $w_F$ in general.
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. Bollobás 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 $\frac 32$.