Related papers: Supersaturation Problem for Color-Critical Graphs

A graph $H$ is $K_s$-saturated if it is a maximal $K_s$-free graph, i.e., $H$ contains no clique on $s$ vertices, but the addition of any missing edge creates one. The minimum number of edges in a $K_s$-saturated graph was determined over…

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 $H$…

In 1975, P. Erd\H{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on…

For an $r$-uniform hypergraph $H$ and a family of $r$-uniform hypergraphs $\mathcal{F}$, the relative Tur\'{a}n number $\mathrm{ex}(H,\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $H$. In this paper we…

Given a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a {\em Berge-$F$} if there is a bijection $\phi:E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. Given a family $\mathcal{F}$ of $r$-uniform…

Given a set of graphs $\mathcal{H}$, we say that a graph $G$ is \textit{$\mathcal{H}$-free} if it does not contain any member of $\mathcal{H}$ as a subgraph. Let $\text{ex}(n,\mathcal{H})$ (resp. $\text{ex}_{sp}(n,\mathcal{H})$) denote the…

A classical Tur\'an problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph $H$ as a subgraph. It is well-known that the chromatic number of $H$ is the graph parameter which…

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+36t$$ for $t=1260r+169…

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 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…

Extremal problems on the $4$-cycle $C_4$ played a heuristic important role in the development of extremal graph theory. A fundamental theorem of F\"uredi states that the Tur\'an number $ex(q^2+q+1, C_4)\leq \frac12 q(q+1)^2$ holds for every…

Let $\mathcal{F}$ be a nonempty family of graphs. A graph $G$ is called $\mathcal{F}$-\textit{free} if it contains no graph from $\mathcal{F}$ as a subgraph. For a positive integer $n$, the \emph{planar Tur\'an number} of $\F$, denoted by…

Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Tur\'an graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large…

For graphs $H$ and $F$, let $\operatorname{ex}(n, H, F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalises the well-studied Tur\'an numbers of graphs, was…

The classical Simonovits' chromatic critical edge theorem shows that for sufficiently large $n$, if $H$ is an edge-color-critical graph with $\chi(H)=p+1\ge 3$, then the Tur\'an graph $T_{n,p}$ is the unique extremal graph with respect to…

Given a graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of an edge set containing at most one edge from each triangle of $G$.…

An edge-coloring of a graph $H$ is a function $\mathcal{C}: E(H) \rightarrow \mathbb{N}$. We say that $H$ is rainbow if all edges of $H$ have different colors. Given a graph $F$, an edge-colored graph $G$ is $F$-rainbow saturated if $G$…

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…

Given a graph $H$ and a set of graphs $\mathcal F$, let $ex(n,H,\mathcal F)$ denote the maximum possible number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. We investigate the function $ex(n,H,\mathcal F)$, when $H$ and…

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. Erd\H{o}s…