SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1165-1181, January 2021.
This paper considers an edge minimization problem in saturated bipartite graphs. An $n$ by $n$ bipartite graph $G$ is $H$-saturated if $G$ does not contain a subgraph isomorphic to $H$ but adding any missing edge to $G$ creates a copy of $H$. More than half a century ago, Wessel and Bollobás independently solved the problem of minimizing the number of edges in $K_{(s,t)}$-saturated graphs, where $K_{(s,t)}$ is the “ordered” complete bipartite graph with $s$ vertices from the first color class and $t$ from the second. However, the very natural “unordered” analogue of this problem was considered only half a decade ago by Moshkovitz and Shapira. When $s=t$, it can be easily checked that the unordered variant is exactly the same as the ordered case. Later, Gan, Korándi, and Sudakov gave an asymptotically tight bound on the minimum number of edges in $K_{(s,t)}$-saturated $n$ by $n$ bipartite graphs, which is only smaller than the conjecture of Moshkovitz and Shapira by an additive constant. In this paper, we confirm their conjecture for $s=t-1$ with the classification of the extremal graphs. We also improve the estimates of Gan, Korándi, and Sudakov for general $s$ and $t$, and for all sufficiently large $n$.

中文翻译：

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最小化 $K_{(s,t)}$-饱和二部图中的边数

SIAM 离散数学杂志，第 35 卷，第 2 期，第 1165-1181 页，2021 年 1 月。
本文考虑了饱和二部图中的边最小化问题。如果 $G$ 不包含与 $H$ 同构的子图但向 $G$ 添加任何缺失的边会创建 $H$ 的副本，则 $n$ 乘 $n$ 二部图 $G$ 是 $H$ 饱和的。半个多世纪前，Wessel 和 Bollobás 独立解决了在 $K_{(s,t)}$ 饱和图中最小化边数的问题，其中 $K_{(s,t)}$ 是“有序”完整的二部图，其中 $s$ 顶点来自第一个颜色类，$t$ 来自第二个颜色类。然而，这个问题的非常自然的“无序”类似物仅在五年前被 Moshkovitz 和 Shapira 考虑过。当 $s=t$ 时，可以很容易地检查无序变体与有序情况完全相同。后来，甘、科兰迪、Sudakov 对 $K_{(s,t)}$-saturated $n$ by $n$ 二部图中的最小边数给出了渐近紧界，仅比 Moshkovitz 和 Shapira 的加法猜想小持续的。在本文中，我们通过极值图的分类证实了他们对 $s=t-1$ 的猜想。我们还改进了 Gan、Korándi 和 Sudakov 对一般 $s$ 和 $t$ 以及所有足够大的 $n$ 的估计。