退化Turán问题中的度幂
On degree powers in the degenerate Turán problem
AI总结：
本文针对退化Turán问题，基于前人方法得到$ex_{p}(n, \mathcal{F})$的稳定性结果，证明$n$足够大时极值图含指定完全二分图，推导二分图$F$的已知结果并得到多类$F$的新精确结果。
AI中文摘要：
给定一个度序列为$d_1,\ldots,d_n$的图$G$和正实数$p$，令$e_p(G)=\sum_{i=1}^{n} d_{i}^{p}$。对于固定的图族$\mathcal{F}$，令$ex_{p}(n, \mathcal{F})$表示$n$个顶点的所有不含$\mathcal{F}$中图的图$G$中$e_p(G)$的最大值。2000年，Caro和Yuster提出了如下Turán型问题：对于正整数$p$和固定图$F$，确定$ex_{p}(n, F)$，并刻画达到该值的$n$顶点极值图$G$。最近，Gao、Liu、Ma和Pikhurko证明，当$\mathcal{F}$是退化图族，且其经典Turán数$ex(n, \mathcal{F})=O(n^{1+\alpha})$（其中$\alpha\in[0,1)$），$\tau(\mathcal{F})$是$\mathcal{F}$中所有二分图的最小独立顶点覆盖大小，且实$p>\frac{1}{1-\alpha}$时，$ex_{p}(n, \mathcal{F})=(\tau(\mathcal{F})-1+o(1))n^p$。基于他们的方法，本文得到$ex_{p}(n, \mathcal{F})$的稳定性结果，并证明当$n$足够大时，所有极值图必须包含完全二分图$K_{\tau(\mathcal{F})-1,n-\tau(\mathcal{F})+1}$。本文结果可推导出当$F$为二分图且$n$足够大时所有此前关于$ex_{p}(n, F)$的已知结果，还针对$F$为偶环、完全二分图、离散超立方体、毛虫森林和蜘蛛森林的情况得到了若干新的精确结果。
英文摘要：
Given a graph $G$ with degree sequence $d_{1},\ldots,d_{n}$ and a positive real number $p$, let $e_{p}(G)=\sum_{i=1}^{n} d_{i}^{p}$. For a fixed family of graphs $\mathcal F$, let $ex_{p}(n, \mathcal F)$ denote the maximum value of $e_{p}(G)$ over all $\mathcal F$-free graphs $G$ on $n$ vertices. In 2000, Caro and Yuster introduced the following Turán-type problem: For a positive integer $p$ and a fixed graph $F$, determine $ex_{p}(n, F)$, and characterize the extremal graphs $G$ on $n$ vertices that attain $ex_p(n, F)$. Recently, Gao, Liu, Ma and Pikhurko proved that $ex_{p}(n, \mathcal F)=(τ(\mathcal F)-1+o(1))n^p$ for real $p>\frac{1}{1-α}$, where $\mathcal F$ is a degenerate family of graphs with classical Turán number $ex(n, \mathcal F)=O(n^{1+α})$ for some $α\in[0,1)$, and $τ(\mathcal F)$ is the minimum size of an independent vertex cover over all bipartite graphs $F\in\mathcal F$. Based on their method, we obtain a stability result for $ex_{p}(n, \mathcal F)$, and prove that all extremal graphs must contain the complete bipartite graph $K_{τ(\mathcal F)-1,n-τ(\mathcal F)+1}$ when $n$ is sufficiently large. Our results can be used to deduce all previously known results about $ex_{p}(n, F)$ when $F$ is a bipartite graph and $n$ is sufficiently large. We also obtain several new exact results for $ex_{p}(n, F)$, namely, when $F$ is an even cycle, a complete bipartite graph, a discrete hypercube, a caterpillar forest, and a spider forest.
