Related papers: Planar Tur\'{a}n Number of intersecting triangles

An $r$-uniform hypergraph is called an $r$-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear $r$-graph $H$ and a positive integer $n$, the linear Tur\'an number $ex_L(n,H)$ is the maximum…

Let $H_{s,t_1,\ldots ,t_k}$ be the graph with $s$ triangles and $k$ odd cycles of lengths $t_1,\ldots ,t_k\ge 5$ intersecting in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126--137], and Yuan [J. Graph…

Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Tur\'an number of Berge-$\mathcal{F}$ is the maximum…

An edge-colored graph $F$ is {\it rainbow} if each edge of $F$ has a unique color. The {\it rainbow Tur\'an number} $\mathrm{ex}^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with…

We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.

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…

The generalized Tur\'{a}n number $\mathrm{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. For an integer $t \geq 1$, let $tF$ be the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku,…

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, and let $H$ be an $r$-uniform hypergraph. Then $H$ is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Tur\'{a}n number of…

We consider a problem proposed by Linial and Wilf to determine the structure of graphs that allows the maximum number of $q$-colorings among graphs with $n$ vertices and $m$ edges. Let $T_r(n)$ denote the Tur\'{a}n graph - the complete…

A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges…

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing…

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…

The generalized Tur\'an number $\text{ex}(n,H,\mathcal{F})$ denotes the maximum number of copies of $H$ in an $n$-vertex graph which contains no copies of any graph in a family $\mathcal{F}$ of graphs. The generalized rational exponents…

Let $P_4$ denote the path graph on $4$ vertices. The suspension of $P_4$, denoted by $\widehat P_4$, is the graph obtained via adding an extra vertex and joining it to all four vertices of $P_4$. In this note, we demonstrate that for $n\ge…

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 $H$…

For positive integers $n$ and $r$, we consider $n$-vertex graphs with the maximum number of $r$-edge-colorings with no copy of a triangle where exactly two colors appear. We prove that, if $2 \leq r \leq 26$ and $n$ is sufficiently large,…

In this paper, we investigate the Tur\'an exponent for $1$-subdivisions of graphs that are neither bipartite nor complete. Specifically, we establish an upper bound on the Tur\'an number of the 1-subdivision of $K_{s,t}^+$, where…

The $r$-expansion $G^+$ of a graph $G$ is the $r$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a vertex subset of size $r-2$ disjoint from $V(G)$ such that distinct edges are enlarged by disjoint subsets. Let…

Let $H$ be a fixed graph. Denote $f(n,H)$ to be the maximum number of edges not contained in any monochromatic copy of $H$ in a 2-edge-coloring of the complete graph $K_n$, and $ex(n,H)$ to be the {\it Tur\'an number} of $H$. An easy lower…

Let $\mathcal{F}$ be a family of $r$-graphs. The Tur\'an number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erd\H{o}s Matching Conjecture shows that…