论文标题
线性森林的广义Turán数字的稳定性
Stability of generalized Turán number for linear forests
论文作者
论文摘要
给定图形$ t $和一个图$ \ MATHCAL {F} $，$ \ MATHCAL {F} $的概括是$ \ Mathcal {f} $中$ t $的最大副本数量 - $ n $ vertices上的免费图形，在$ n $ vertices上，$ n $ vertices，$ n $ vertices，由$ ex（n，t，t，t，t，mathcal c}表示）。当$ t = k_r $时，$ ex（n，k_r，\ mathcal {f}）$是一个函数，在$ \ mathcal {f} $中指定最大数量的$ r $ -cliques，$ n $ dertices上的免费图。线性森林是一片森林，其连接成分都是路径和孤立的顶点。令$ \ Mathcal {l} _ {k} $为所有尺寸$ k $的所有线性森林的家族，而无需隔离顶点。在本文中，我们在$ g $中获得了最大可能的$ r $ cliques数量，其中$ g $是$ \ mathcal {l} _ {k} $ - 至少至少$ d $免费。此外，我们给出了结果的稳定版本。作为结果的稳定版本的应用，我们获得了匹配中Erdős-Gallai定理的稳定性集团版本。
Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Turán number of $\mathcal{F}$ is the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices, denoted by $ex(n,T,\mathcal{F})$. When $T = K_r$, $ex(n, K_r, \mathcal{F})$ is a function specifying the maximum possible number of $r$-cliques in an $\mathcal{F}$-free graph on $n$ vertices. A linear forest is a forest whose connected components are all paths and isolated vertices. Let $\mathcal{L}_{k}$ be the family of all linear forests of size $k$ without isolated vertices. In this paper, we obtained the maximum possible number of $r$-cliques in $G$, where $G$ is $\mathcal{L}_{k}$-free with minimum degree at least $d$. Furthermore, we give a stability version of the result. As an application of the stability version of the result, we obtain a clique version of the stability of the Erdős-Gallai Theorem on matchings.
