Weak Saturation in Graphs: a Combinatorial Approach

The weak saturation number $\mathrm{wsat}(n,F)$ is the minimum number of edges in a graph on $n$ vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of $F$. A usual approach to prove a lower bound for the weak saturation number is algebraic: if it is possible to embed edges of $K_n$ in a vector space in a certain way (depending on $F$), then the dimension of the subspace spanned by the images of the edges of $K_n$ is a lower bound for the weak saturation number. In this paper, we present a new combinatorial approach to prove lower bounds for weak saturation numbers that allows to establish worst-case tight (up to constant additive terms) general lower bounds as well as to get exact values of the weak saturation numbers for certain graph families. It is known (Alon, 1985) that, for every $F$, there exists $c_F$ such that $\mathrm{wsat}(n,F)=c_Fn(1+o(1))$. Our lower bounds imply that all values in the interval $\left[\frac{\delta}{2}-\frac{1}{\delta+1},\delta-1\right]$ with step size $\frac{1}{\delta+1}$ are achievable by $c_F$ (while any value outside this interval is not achievable).