arXiv 1402.24712014-02math.CO

$K_{s,t}$ -饱和二部图

$K_{s,t}$-saturated bipartite graphs

Wenying Gan, Dániel Korándi, Benny Sudakov

备注：: 11 pages, 1 figure

摘要：

一个$n$乘$n$二部图是$H$饱和的，如果在它的两个部分之间添加任何缺失的边都会创建一个新的$H$副本。1964年，Erdés，Hajnal和Moon提出了关于$K_{s,s}$ -饱和二部图的最小边数的猜想.这个猜想由Wessel和Bollobás在一个更一般但有序的环境中独立证明：他们证明了$K_{(s,t)}$饱和二部图的最小边数是$n^2-(n-s+1)(n-t+1)$，其中$K_{(s,t)}$是“有序”完全二部图，其中$s$顶点在第一个颜色类，$t$顶点在第二个颜色类。然而，在无序$K_{s,t}$饱和的情况下确定最小边数的非常自然的问题仍然没有解决。这个问题最近被Moshkovitz和Shapira考虑过，他们也证明了它的答案应该是什么。本文给出了$K_{s,t}$ -饱和二部图的最小边数的一个渐近紧界，它仅比Moshkovitz和Shapira的猜想小一个加性常数.我们还证明了他们的猜想$K_{2,3}$ -饱和，这是第一个开放的情况。

英文摘要：

An $n$-by-$n$ bipartite graph is $H$-saturated if the addition of any missing edge between its two parts creates a new copy of $H$. In 1964, Erdős, Hajnal and Moon made a conjecture on the minimum number of edges in a $K_{s,s}$-saturated bipartite graph. This conjecture was proved independently by Wessel and Bollobás in a more general, but ordered, setting: they showed that the minimum number of edges in a $K_{(s,t)}$-saturated bipartite graph is $n^2-(n-s+1)(n-t+1)$, where $K_{(s,t)}$ is the "ordered" complete bipartite graph with $s$ vertices in the first color class and $t$ vertices in the second. However, the very natural question of determining the minimum number of edges in the unordered $K_{s,t}$-saturated case remained unsolved. This problem was considered recently by Moshkovitz and Shapira who also conjectured what its answer should be. In this short paper we give an asymptotically tight bound on the minimum number of edges in a $K_{s,t}$-saturated bipartite graph, which is only smaller by an additive constant than the conjecture of Moshkovitz and Shapira. We also prove their conjecture for $K_{2,3}$-saturation, which was the first open case.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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