Weak saturation numbers of large complete bipartite graphs

TL;DR

This paper determines exact weak saturation numbers for large complete bipartite graphs in a new parameter range, introducing a novel method involving hypergraph auxiliary structures and connectivity properties.

Contribution

It provides exact values and bounds for _{s,t}(n) in previously unaddressed ranges, using a new approach with hypergraph and connectivity analysis.

Findings

Exact values for _{s,t}(n) in new range

A new method involving auxiliary hypergraphs

Tight bounds up to an additive constant

Abstract

An $n$-vertex graph $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ and there exists an ordering of all edges in $E(K_n) \setminus E(G)$ such that, when added one at a time, each edge creates a new copy of $F$. The minimum size of a weakly $F$-saturated graph $G$ is called the weak saturation number $\mathrm{wsat}(n, F)$. We obtain exact values and new bounds for $\mathrm{wsat}(n, K_{s,t})$ in the previously unaddressed range $s+t < n < 3t-3$, where $3\leq s\leq t$. To prove lower bounds, we introduce a new method that takes into account connectivity properties of subgraphs of a complement $G'$ to a weakly saturated graph $G$. We construct an auxiliary hypergraph and show that a linear combination of its parameters always increases in the process of the deletion of edges of $G'$. This gives a lower bound which is tight, up to an additive constant.