Georgia Institute of Technology College of Sciences

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 $wsat(n,F)$ 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 $wsat(n,F) = w_F n + o(n)$. We characterize all possible rational values of $w_F$, proving in particular that $w_F$ can equal any rational number at least $3/2$. The techniques involve a combination of random and deterministic constructions and structural methods. Joint work with Xiaoyu He.