This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. We first resolve a fundamental question of minimizing the number of cliques of size $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and Timmons. We also go further and prove a corresponding stability result. Next we minimize the number of cycles of length $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, and classify the extremal graphs for most values of $r$, answering another question of Kritschgau, Methuku, Tait, and Timmons for most $r$.

We then move on to a central and longstanding conjecture in graph saturation made by Tuza, which states that for every graph $F$, the limit ${lim}_{n\to \infty }\frac{sat(n,F)}{n}$ exists, where $sat(n,F)$ denotes the minimum number of edges in an $n$-vertex $F$-saturated graph. Pikhurko made progress in the negative direction by considering families of graphs instead of a single graph, and proved that there exists a graph family $\mathcal{F}$ of size 4 for which ${lim}_{n\to \infty }\frac{sat(n,\mathcal{F})}{n}$ does not exist (for a family of graphs $\mathcal{F}$, a graph $G$ is called $\mathcal{F}$-saturated if $G$ does not contain a copy of any graph in $\mathcal{F}$, but the addition of any edge creates a copy of a graph in $\mathcal{F}$, and $sat(n,\mathcal{F})$ is defined similarly). We make the first improvement in 15 years by showing that there exist infinitely many graph families of size 3 where this limit does not exist. Our construction also extends to the generalized saturation problem when we minimize the number of fixed-size cliques. We also show an example of a graph $F_r$ for which there is irregular behavior in the minimum number of $C_r$’s in an $n$-vertex $F_r$-saturated graph.

本文考虑了精心研究的图论中的两个重要问题： $F$-饱和。图$G$ 叫做 $F$-饱和如果 $G$ 不包含同构的子图 $F$，但添加任何边都会创建的副本 $F$。我们首先解决一个基本问题，即最大程度地减少规模集团$\mathrm{[R}$ 在一个 ${ķ}_s$-所有足够多的顶点的饱和图，确认了Kritschgau，Methuku，Tait和Timmons的猜想。我们还进一步证明了相应的稳定性结果。接下来，我们最小化长度的循环数$\mathrm{[R}$ 在一个 ${ķ}_s$-饱和图，用于所有足够多的顶点，并对最大值的极值图进行分类 $\mathrm{[R}$，在大多数情况下回答Kritschgau，Methuku，Tait和Timmons的另一个问题 $\mathrm{[R}$。

然后，我们继续讨论由图扎（Tuza）进行的图饱和的中心且长期存在的猜想，其中指出了每个图 $F$， 极限 ${林}_{ñ\to \infty }\frac{坐着（ñ，F）}{ñ}$ 存在，在哪里 $坐着（ñ，F）$ 表示一个 $ñ$-顶点 $F$-饱和图。Pikhurko通过考虑图族而不是单个图在负向方面取得了进步，并证明存在图族$\mathcal{F}$ 尺寸4 ${林}_{ñ\to \infty }\frac{坐着（ñ，\mathcal{F}）}{ñ}$ 不存在（对于一系列图 $\mathcal{F}$图 $G$ 叫做 $\mathcal{F}$-饱和如果 $G$ 不包含任何图形的副本 $\mathcal{F}$，但是任何边的添加都会在中创建图的副本 $\mathcal{F}$和 $坐着（ñ，\mathcal{F}）$的定义与此类似）。通过显示不存在此限制的无限大的3号图形族，我们实现了15年中的第一个改进。当我们最小化固定大小的团体时，我们的构造还扩展到广义饱和问题。我们还展示了一个图形示例$F_{\mathrm{[R}}$ 其最小数量的 $C_{\mathrm{[R}}$在 $ñ$-顶点 $F_{\mathrm{[R}}$-饱和图。