Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. K\'aszonyi and Tuza, in 1986, gave a general upper bound on the saturation number of a graph $H$, but a nontrivial lower bound has remained elusive. In this paper we give a general lower bound on $\mathrm{sat}(H,n)$ and prove that it is asymptotically sharp (up to an additive constant) on a large class of graphs. This class includes all threshold graphs and many graphs for which the saturation number was previously determined exactly. Our work thus gives an asymptotic common generalization of several earlier results. The class also includes disjoint unions of cliques, allowing us to address an open problem of Faudree, Ferrara, Gould, and Jacobson.
翻译:$H 是一个固定的图形 。 我们说, 图表 $G$ 是 $H$ 饱和的 $H$, 如果它没有子图是 $H 的, 但是将任何边缘加到 $G$ 等于 $H$ 的 美元 。 饱和的 $( H, n) $ 是 $H$ 饱和的 图 的最小边数 。 1986 年 K\ aszonyi 和 Tuza 给出了 $H$ 的饱和度总上限, 但是一个非三维的下限仍然难以找到。 在本文中, 我们给$\ mathrm{sat} ( H, n) 以一般下限为 $\ mathrm{ sat} ( H, n), 并证明它是一个大类图中的“ $H$- 饱和 常数 ” 。 这个类包含所有 门槛的图形和许多图表, 先前确定饱和图数 。 我们的工作因此给出了一些 开放式的通用的通用通则 。 。, 类中包括了 Goqol 和 。