论文标题
$ \ boldsymbol的弱饱和数{k_ {2，t}} $
The weak saturation number of $\boldsymbol{K_{2, t}}$
论文作者
论文摘要
对于两个图表$ g $和$ f $，我们说$ g $是薄弱的$ f $ - 饱和，如果$ g $不包含$ f $作为子图的副本，并且可以加入所有$ g $的所有非附上$ g $的顶点，以便在每个步骤中创建$ f $的新副本。弱饱和数$ \ mathrm {wsat}（n，f）$是$ n $ vertices上弱$ f $饱和图的最小边数。在本文中，我们检查了$ \ mathrm {wsat}（n，k_ {s，t}）$，其中$ k_ {s，t} $是完整的两部分图，带有尺寸$ s $ s $和$ t $的零件。 我们确定$ \ mathrm {wsat}（n，k_ {2，t}）$，纠正文献中的先前报告。还显示出$ \ mathrm {wsat}（s+t，k_ {s，t}）= \ binom {s+t-1} {2} {2} $如果$ \ gcd（s，s，t）= 1 $ and $ \ mathrm {wsat}（wsat}（s+t，s+t，s+t，s+t，s+t，s+t} k_ {s，t}）= \ binom {s+t-1} {2}+1 $，否则。
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.
