标题： 局部团界在度数有界和路径有界图中的研究 显示英文标题

标题： Localized Clique Bounds in Bounded-Degree and Bounded-Path Graphs

摘要： 设$\mathcal{F}$是一个图的族。 如果一个图不包含$\mathcal{F}$中的任何成员，则称该图为$\mathcal{F}$-free。 广义的Turán数$ex(n,H,\mathcal{F})$表示在具有$n$个顶点的$\mathcal{F}$-自由图中，图$H$的副本的最大数量，而广义的边Turán数$mex(m,H,\mathcal{F})$表示在具有$m$条边的$\mathcal{F}$-自由图中，图$H$的副本的最大数量。 已知如果一个图的最大度数为$d$，则它是$K_{1,d+1}$-free。Wood\cite{wood}证明了$ex(n,K_t,K_{1,d+1}) \leq \frac{n}{d+1}\binom{d+1}{t}$。最近，Chakraborty 和 Chen\cite{CHAKRABORTI2024103955}建立了最大路径长度有界的图的类似界限：$mex(m,K_t,P_{r+1}) \leq \frac{m}{\binom{r}{2}}\binom{r}{t}$。在本文中，我们使用基于适当定义的局部参数的定位技术来改进这些界限。此外，我们表征了达到这些改进界限的极值图。 显示英文摘要

摘要： Let $\mathcal{F}$ be a family of graphs. A graph is said to be $\mathcal{F}$-free if it contains no member of $\mathcal{F}$. The generalized Tur\'{a}n number $ex(n,H,\mathcal{F})$ denotes the maximum number of copies of a graph $H$ in an $n$-vertex $\mathcal{F}$-free graph, while the generalized edge Tur\'{a}n number $mex(m,H,\mathcal{F})$ denotes the maximum number of copies of $H$ in an $m$-edge $\mathcal{F}$-free graph. It is well known that if a graph has maximum degree $d$, then it is $K_{1,d+1}$-free. Wood \cite{wood} proved that $ex(n,K_t,K_{1,d+1}) \leq \frac{n}{d+1}\binom{d+1}{t}$. More recently, Chakraborty and Chen \cite{CHAKRABORTI2024103955} established analogous bounds for graphs with bounded maximum path length: $mex(m,K_t,P_{r+1}) \leq \frac{m}{\binom{r}{2}}\binom{r}{t}$. In this paper, we improve these bounds using the localization technique, based on suitably defined local parameters. Furthermore, we characterize the extremal graphs attaining these improved bounds.