Related papers: Some exact results for generalized Tur\'an problem…

For a graph $H$ and a $k$-chromatic graph $F,$ if the Tur\'an 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 $F$-Tur\'an-good, or $k$-Tur\'an-good…

For graphs $H$ and $F$ with chromatic number $\chi(F)=k$, we call $H$ strictly $F$-Tur\'an-good (or $(H, F)$ strictly Tur\'an-good) if the Tur\'an graph $T_{k-1}(n)$ is the unique $F$-free graph on $n$ vertices containing the largest number…

For graphs $H$ and $F$, the generalized Tur\'an number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We say that $H$ is $F$-Tur\'an-good if $ex(n,H,F)$ is the number of copies in the…

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 general Tur\'an number, denoted by $ex(n, H,\mathscr{F})$, is the maximum number of copies of $H$…

Given two graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in an $n$-vertex $F$-free graph. For every $F$ and sufficiently large $n$, we present an extremal graph for a…

The generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $\chi(H)<\chi(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the…

The generalized Tur\'an problem $ext(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Tur\'an…

We show that among $K_{k+1}$-free $n$-vertex graphs, the Tur\'an graph contains the most copies of any path.

We are given graphs $H_1,\dots,H_k$ and $F$. Consider an $F$-free graph $G$ on $n$ vertices. What is the largest sum of the number of copies of $H_i$? The case $k=1$ has attracted a lot of attention. We also consider a colored variant,…

In a generalized Tur\'an problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $F$-free graph of order $n$. We consider generalized Tur\'an problems where the host graph is planar. In particular we…

For fixed graphs $F$ and $H$, the generalized Tur\'an problem asks for the maximum number $ex(n,H,F)$ of copies of $H$ that an $n$-vertex $F$-free graph can have. In this paper, we focus on cases with $F$ being $B_{r,s}$, the graph…

In the so-called generalized Tur\'an problems we study the largest number of copies of $H$ in an $n$-vertex $F$-free graph $G$. Here we introduce a variant, where $F$ is not forbidden, but we restrict how copies of $H$ and $F$ can be placed…

Given graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. We study the case when either $H$ or $F$ is a matching. We obtain several asymptotic and…

Given an $(r+1)$-chromatic graph $F$ and a graph $H$ that does not contain $F$ as a subgraph, we say that $H$ is strictly $F$-Tur\'an-good if the Tur\'an graph $T_{r}(n)$ is the unique graph containing the maximum number of copies of $H$…

Given a graph $H$ and a family of graphs $\mathcal{F}$, the generalized Tur\'an number $\mathrm{ex}(n,H,\mathcal{F})$ is the maximum number of copies of $H$ in an $n$-vertex graphs that do not contain any member of $\mathcal{F}$ as a…

For graphs $H$ and $F$, the generalized Tur\'an number $ex(n,H,F)$ is the largest number of copies of $H$ in an $F$-free graph on $n$ vertices. We consider this problem when both $H$ and $F$ have at most four vertices. We give sharp results…

Given graphs $H$ and $F$, $\mathrm{ex}(n,H,F)$ denotes the largest number of copies of $H$ in $F$-free $n$-vertex graphs. Let $\chi(H)<\chi(F)=r+1$. We say that $H$ is $F$-Tur\'an-stable if the following holds. For any $\varepsilon>0$ there…

Given a graph $T$ and a family of graphs $\mathcal{H}$. The generalized Tur\'an number of $\mathcal{H}$ is the maximum number of copies of $T$ in an $\mathcal{H}$-free graph on $n$ vertices, denoted by $ex(n, T, \mathcal{H})$. Let $ex(n, T,…

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…

We combine two generalizations of ordinary Tur\'an problems. Given graphs $H$ and $F$ and a positive integer $n$, we study $rex(n, H, F )$, which is the largest number of copies of $H$ in $F$-free regular $n$-vertex graphs.