Degree powers in graphs with a forbidden forest
| Autor: | Lan, Yongxin, Liu, Henry, Qin, Zhongmei, Shi, Yongtang |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Given a positive integer $p$ and a graph $G$ with degree sequence $d_1,\dots,d_n$, we define $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: Given a positive integer $p$ and a graph $H$, determine the function $ex_p(n,H)$, which is the maximum value of $e_p(G)$ taken over all graphs $G$ on $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Tur\'an number. Caro and Yuster determined the function $ex_p(n, P_\ell)$ for sufficiently large $n$, where $p\geq 2$ and $P_\ell$ denotes the path on $\ell$ vertices. In this paper, we generalise this result and determine $ex_p(n,F)$ for sufficiently large $n$, where $p\geq 2$ and $F$ is a linear forest. We also determine $ex_p(n,S)$, where $S$ is a star forest; and $ex_p(n,B)$, where $B$ is a broom graph with diameter at most six. Comment: 24 pages, 2 figures |
| Databáze: | arXiv |
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