Related papers: Vertex-Based Localization of Tur\'{a}n's Theorem

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 generalized Tur\'an number $\ex(n,K_s,F)$ denotes the maximum number of copies of $K_s$ in an $n$-vertex $F$-free graph. Let $kF$ denote $k$ disjoint copies of $F$. Gerbner, Methuku and Vizer [DM, 2019, 3130-3141] gave a lower bound for…

The classical Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Theorem states that for $\ell\ge 2$, every $n$-vertex $K_{\ell+1}$-free graph with minimum degree greater than $\frac{3\ell-4}{3\ell-1}n$ must be $\ell$-partite. We establish a simple…

The generalized Tur\'an number $\text{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Let $kK_{r+1}$ be the disjoint union of $k$ copies of the complete graph $K_{r+1}$. Recently, Gerbner…

The Tur\'{a}n number $T(n,\alpha+1,r)$ is the minimum number of edges in an $n$-vertex $r$-graph whose independence number does not exceed $\alpha$. For each $r\geq 2$, there exists $t_*(r)$ such that $T(n,\alpha+1,r) = t_*(r) \: n^r \:…

For two $s$-uniform hypergraphs $H$ and $F$, the Tur\'{a}n number $ex_s(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Let $s, r, k, n_1, \ldots, n_r$ be integers satisfying $2\leq s\leq r$ and $n_1\leq n_2\leq…

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…

Let $\mathcal{F}$ denote a set of graphs. A graph $G$ is said to be $\mathcal{F}$-free if it does not contain any element of $\mathcal{F}$ as a subgraph. The Tur\'an number is the maximum possible number of edges in an $\mathcal{F}$-free…

Given a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n, \mathcal{F})$ denotes the maximum number of edges in any $\mathcal{F}$-free graph on $n$ vertices. Recently, Alon and Frankl studied of maximum number of edges in an…

A well-known result in extremal spectral graph theory, due to Nosal and Nikiforov, states that if $G$ is a triangle-free graph on $n$ vertices, then $\lambda (G) \le \lambda (K_{\lfloor \frac{n}{2}\rfloor, \lceil \frac{n}{2} \rceil })$,…

The well-known Tur\'{a}n theorem states that if $G$ is an $n$-vertex $K_{r+1}$-free graph, then $e(G)\le e(T_{n,r})$, with equality if and only if $G$ is the $r$-partite Tur\'{a}n graph $T_{n,r}$. A graph $F$ is called color-critical if it…

Generalized Tur\'an problem with given size, denoted as $\mathrm{mex}(m,K_r,F)$, determines the maximum number of $K_r$-copies in an $F$-free graph with $m$ edges. We prove that for $r\ge 3$ and $\alpha\in(\frac 2 r,1]$, any graph $G$ with…

Given two graphs $T$ and $F$, the maximum number of copies of $T$ in an $F$-free graph on $n$ vertices is called the generalized Tur\'{a}n number, denoted by $ex(n,T,F)$. When $T=K_2$, it reduces to the classical Tur\'{a}n number $ex(n,F)$.…

A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Zhai and Lin [J. Graph Theory 102 (2023) 502-520] proved that for $n\geq\frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then…

We prove that if $G$ is a graph and $f(v) \leq 1/(d(v) + 1/2)$ for each $v\in V(G)$, then either $G$ has an independent set of size at least $\sum_{v\in V(G)}f(v)$ or $G$ contains a clique $K$ such that $\sum_{v\in K}f(v) > 1$. This result…

Given a graph $H$ and a positive integer $n$, the Tur\'{a}n number of $H$ for the order $n$, denoted $ex(n,H)$, is the maximum size of a simple graph of order $n$ not containing $H$ as a subgraph. Given graphs $G$ and $H$, the notation $G…

Zykov's theorem shows that $r$-partite Tur\'{a}n graph uniquely has the maximum number of $K_t$ among all $n$-vertex $K_{r+1}$-free graphs for $2\le t\le r$. The clique tensor is a high-order extension of the adjacency matrix of a graph. Yu…

The spectral Tur\'an theorem states that the $k$-partite Tur\'an graph is the unique graph attaining the maximum adjacency spectral radius among all graphs of order $n$ containing no the complete graph $K_{k+1}$ as a subgraph. This result…

The Tur\'{a}n number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension $\mathcal{F}+v$…

Confirming a conjecture of Elphick and Edwards and strengthening a spectral theorem of Wilf, Nikiforov proved that for any $K_{r+1}$-free graph $G$, $\lambda(G)^2 \leq 2 (1 - 1/r) m$, where $\lambda(G)$ is the spectral radius of $G$, and…