相关论文: A Note on Tur\'an Numbers and the Erd\H{o}s-Stone-…

Given a graph $L$, the Tur\'an number $\textrm{ex}(n,L)$ is the maximum possible number of edges in an $n$-vertex $L$-free graph. The study of Tur\'an number of graphs is a central topic in extremal graph theory. Although the celebrated…

Given a graph $H,$ we say that a graph is \textit{$H$-free} if it does not contain $H$ as a subgraph. The Tur\'an number $\ex(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex $H$-free graph, the set of all the corresponding…

Fix a graph $F$. We say that a graph is {\it $F$-free} if it does not contain $F$ as a subgraph. The {\it Tur\'an number} of $F$, denoted $\mathrm{ex}(n,F)$, is the maximum number of edges possible in an $n$-vertex $F$-free graph. The study…

The extremal number of a graph $H$, denoted by $\mbox{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices that does not contain $H$. The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem says that for a complete bipartite graph…

The Tur\'an number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine…

In this paper we initiate a systematic study of the Tur\'an problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the…

For a family of graphs $\cal F$, a graph $G$ is $\cal F$-free if it does not contain a member of $\cal F$ as a subgraph. The Tur\'an number $\textrm{ex}(n,{\cal F})$ is the maximum number of edges in an $n$-vertex graph which is $\cal…

The classical stability theorem of Erd\H{o}s and Simonovits states that, for any fixed graph with chromatic number $k+1 \ge 3$, the following holds: every $n$-vertex graph that is $H$-free and has within $o(n^2)$ of the maximal possible…

A fundamental problem of extremal graph theory is to ask, 'What is the maximum number of edges in an $F$-free graph on $n$ vertices?' Recently Alon and Shikhelman proposed a more general, subgraph counting, version of this question. They…

The Tur\'an number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. Let P_k be the path with k vertices, the square P^2_k of P_k is obtained by joining the pairs of vertices…

For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $\delta(H)>0$ and $\Delta(H)$ denote the minimum degree and maximum degree of $H$,…

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability Theorem of Erd\H{o}s and Simonovits shows that if a $K_{r+1}$-free graph with $n$ vertices has close to the maximal $t_r(n)$ edges, then it is close…

The Tur\'{a}n number of a graph $H$, denoted $\mbox{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph with no subgraph isomorphic to $H$. Solymosi conjectured that if $H$ is any graph and $\mbox{ex}(n,H) = O(n^{\alpha})$…

The Tur\'an number $\text{ex}(n,H)$ of a graph $H$ is the maximal number of edges in an $H$-free graph on $n$ vertices. In $1983$ Chung and Erd\H{o}s asked which graphs $H$ with $e$ edges minimize $\text{ex}(n,H)$. They resolved this…

The Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free graph on $n$ vertices. Let $T$ be any tree. The odd-ballooning of $T$, denoted by $T_o$, is a graph obtained by replacing each edge of $T$ with an odd cycle…

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathcal{F}$ as a subgraph. The Tur\'an number $ex(n, \mathscr{F})$ is the maximum number of edges in an $n$-vertex…

The cube Q is the usual 8-vertex graph with 12 edges. Here we give a new proof for a theorem of Erd\H{o}s and Simonovits concerning the Tur\'an number of the cube. Namely, it is shown that e(G) < n^{8/5}+(2n)^{3/2} holds for any n-vertex…

The Tur\'an number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in a graph on $n$ vertices which does not have $H$ as a subgraph. A wheel $W_n$ is an $n$-vertex graph formed by connecting a single vertex to all vertices…

Erd\H{o}s-Stone Theorem is a well-known result in extremal graph theory which determines the asymptotic behaviour of maximum number of edges in an $n$-vertex $H$-free graph. In 2009, Nikiforov gave a spectral version of Erd\H{o}s-Stone…

Let $\cal H$ be a family of graphs. The Tur\'an number ${\rm ex}(n,{\cal H})$ is the maximum possible number of edges in an $n$-vertex graph which does not contain any member of $\cal H$ as a subgraph. As a common generalization of…