Weak saturation in graphs: A combinatorial approach

Abstract

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. In contrast to previous algebraic approaches, 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 $\mathrm{wsat}(n,F)=c_{F}n(1+o(1\left)\right)$. Our lower bounds imply that all values in the interval $[\frac{\delta }{2}-\frac{1}{\delta +1},\delta -1]$ with step size $\frac{1}{\delta +1}$ are achievable by $c_F$ for graphs F with minimum degree δ (while any value outside this interval is not achievable).