Related papers: Tur\'an Problems for Mixed Graphs

The classical extremal function for a graph $H$, $ex(K_n, H)$ is the largest number of edges in a subgraph of $K_n$ that contains no subgraph isomorphic to $H$. Note that defining $ex(K_n, H-ind)$ by forbidding induced subgraphs isomorphic…

The Tur\'an number of a graph $H$, denoted by $ex(n, H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let $P_{\ell}$ denote the path on $\ell$ vertices, $S_{\ell-1}$ denote the star on $\ell$…

The \textit{generalized Tur\'an number} $\mathrm{ex}(n, T, F)$ is the maximum possible number of copies of $T$ in an $F$-free graph on $n$ vertices for any two graphs $T$ and $F$. For the book graph $B_t$, there is a close connection…

For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of $\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. The determination of Tur\'{a}n number…

In this paper we continue the study of a natural generalization of Tur\'an's forbidden subgraph problem and the Ruzsa-Szemer\'edi problem. Let $ex_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that…

Let $F$ be a graph. We say that a hypergraph $H$ contains an induced Berge $F$ if the vertices of $F$ can be embedded to $H$ (e.g., $V(F)\subseteq V(H)$) and there exists an injective mapping $f$ from the edges of $F$ to the hyperedges of…

The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there…

Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove…

For two graphs $G$ and $H$, the Tur\'{a}n number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Tur\'{a}n numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this…

For integers $1\le \ell<k$, the $\ell$-degree Tur\'an density $\pi_\ell(F)$ measures the minimum $\ell$-degree threshold that forces a copy of a fixed $k$-uniform hypergraph $F$, generalizing both the classical Tur\'an density $\pi_1$ and…

The uniform Tur\'an density $\pi_{1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining…

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…

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations.…

Let $H_n$ be a $k$-graph on $n$ vertices. For $0 \le \ell <k$ and an $\ell$-subset $T$ of $V(H_n)$, define the degree $\deg(T)$ of $T$ to be the number of $(k-\ell)$-subsets~$S$ such that $S \cup T$ is an edge in~$H_n$. Let the minimum…

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…

Given a family of $r$-uniform hypergraphs ${\cal F}$ (or $r$-graphs for brevity), the Tur\'an number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain any member of ${\cal…

Generalized Tur\'an problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain…

The Tur\'an number of a graph $H$, denoted by $ex(n, H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let $P_k$ denote the path on $k$ vertices, $S_k$ denote the star on $k+1$ vertices and…

Let $\mathrm{ex}(n, H)$ be the Tur\'{a}n number of $H$ for a given graph $H$. A graph is color-critical if it contains an edge whose removal reduces its chromatic number. Simonovits' chromatic critical edge theorem states that if $H$ is…

We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges…