ﻻ يوجد ملخص باللغة العربية
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 following problem in 1990: For which graphs $H$ is it true that every graph on $n$ vertices and ${rm ex}(n,H)+1$ edges contains at least two $H$s? Perhaps this is always true. We solve the second part of this problem in the negative by proving that for every integer $kge 4,$ there exists a graph $H$ of order $k$ and at least two orders $n$ such that there exists a graph of order $n$ and size ${rm ex}(n,H)+1$ which contains exactly one copy of $H.$ Denote by $C_4$ the $4$-cycle. We also prove that for every integer $n$ with $6le nle 11,$ there exists a graph of order $n$ and size ${rm ex}(n,C_4)+1$ which contains exactly one copy of $C_4,$ but for $n=12$ or $n=13,$ the minimum number of copies of $C_4$ in a graph of order $n$ and size ${rm ex}(n,C_4)+1$ is $2.$
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
The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph $G$ with $lfloor n^2/4 rfloor +t$ edges and triangle covering number $s$, we determine (for large
Given positive integers $p$ and $q$, a $(p,q)$-coloring of the complete graph $K_n$ is an edge-coloring in which every $p$-clique receives at least $q$ colors. ErdH{o}s and Shelah posed the question of determining $f(n,p,q)$, the minimum number of co
A graph is $P_8$-free if it contains no induced subgraph isomorphic to the path $P_8$ on eight vertices. In 1995, ErdH{o}s and Gy{a}rf{a}s conjectured that every graph of minimum degree at least three contains a cycle whose length is a power of two.
Given a sequence $mathbf{k} := (k_1,ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,dots,s$ such that, for every $c in {1,dots,s}$, the edges of colour $c$ conta