The weak saturation number of $\boldsymbol{K_{2, t}}$

TL;DR

This paper determines the weak saturation number for the bipartite graph $K_{2,t}$ and explores the weak saturation number for $K_{s,t}$, revealing exact formulas based on the greatest common divisor of $s$ and $t$.

Contribution

It provides the exact value of the weak saturation number for $K_{2,t}$ and clarifies the formula for $K_{s,t}$ depending on $ ext{gcd}(s,t)$, correcting previous literature.

Findings

$ ext{wsat}(n, K_{2,t})$ is explicitly determined.

$ ext{wsat}(s+t, K_{s,t})$ equals $inom{s+t-1}{2}$ if $ ext{gcd}(s,t)=1$.

$ ext{wsat}(s+t, K_{s,t})$ equals $inom{s+t-1}{2}+1$ otherwise.

Abstract

For two graphs $G$ and $F$, we say that $G$ is weakly $F$-saturated if $G$ contains no copy of $F$ as a subgraph and one could join all the nonadjacent pairs of vertices of $G$ in some order so that a new copy of $F$ is created at each step. The weak saturation number $\mathrm{wsat}(n, F)$ is the minimum number of edges of a weakly $F$-saturated graph on $n$ vertices. In this paper, we examine $\mathrm{wsat}(n, K_{s, t})$, where $K_{s, t}$ is the complete bipartite graph with parts of sizes $s$ and $ t $. We determine $\mathrm{wsat}(n, K_{2, t})$, correcting a previous report in the literature. It is also shown that $\mathrm{wsat}(s+t, K_{s,t})=\binom{s+t-1}{2}$ if $\gcd(s, t)=1$ and $\mathrm{wsat}(s+t, K_{s,t})=\binom{s+t-1}{2}+1$, otherwise.