ﻻ يوجد ملخص باللغة العربية
The generalized Tur{a}n number $ex(n,K_s,H)$ is defined to be the maximum number of copies of a complete graph $K_s$ in any $H$-free graph on $n$ vertices. Let $F$ be a linear forest consisting of $k$ paths of orders $ell_1,ell_2,...,ell_k$. In this paper, by characterizing the structure of the $F$-free graph with large minimum degree, we determine the value of $ex(n,K_s,F)$ for $n=Omegaleft(|F|^sright)$ and $kgeq 2$ except some $ell_i=3$, and the corresponding extremal graphs. The special case when $s=2$ of our result improves some results of Bushaw and Kettle (2011) and Lidick{y} et al. (2013) on the classical Tur{a}n number for linear forests.
Let $mathcal{F}$ be a family of graphs. A graph $G$ is called textit{$mathcal{F}$-free} if for any $Fin mathcal{F}$, there is no subgraph of $G$ isomorphic to $F$. Given a graph $T$ and a family of graphs $mathcal{F}$, the generalized Tur{a}n number
Given a graph $H$ and a positive integer $n,$ the Tur{a}n number of $H$ for the order $n,$ denoted ${rm ex}(n,H),$ is the maximum size of a simple graph of order $n$ not containing $H$ as a subgraph. The book with $p$ pages, denoted $B_p$, is the gra
For a graph $H$ and a $k$-chromatic graph $F,$ if the Turan graph $T_{k-1}(n)$ has the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough), then $H$ is called $F$-Turan-good, or $k$-Turan-good for short if $F$
We consider finite simple graphs. Given a graph $H$ and a positive integer $n,$ the Tur{a}n number of $H$ for the order $n,$ denoted ${rm ex}(n,H),$ is the maximum size of a graph of order $n$ not containing $H$ as a subgraph. ErdH{o}s posed the foll
While the game chromatic number of a forest is known to be at most 4, no simple criteria are known for determining the game chromatic number of a forest. We first state necessary and sufficient conditions for forests with game chromatic number 2 and