Related papers: Extremal results for $\mathcal{K}^-_{r + 1}$-free …

Recently, Alon and Frankl (JCTB, 2024) determined the maximum number of edges in $K_{\ell+1}$-free $n$-vertex graphs with bounded matching number. For integers $\ell\ge r \ge 2$, the family $\mathcal{K}_{\ell+1}^{r}$ consists of all…

An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containing any member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ be the collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that…

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…

For a simple graph $F$, let $\mathrm{EX}(n, F)$ and $\mathrm{EX_{sp}}(n,F)$ be the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph…

In this paper, we study the problem of determining the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that contains no $(k+1)$-connected subgraph. The graph case is a classical problem initiated by Mader, central to graph…

For a family of graphs $\F$, a graph is called $\F$-free if it does not contain any member of $\F$ as a subgraph. The generalized Tur\'an number $\ex(n,K_r,\F)$ is the maximum number of $K_r$ in an $n$-vertex $\F$-free graph and…

We study the behaviour of $K_{r+1}$-free graphs $G$ of almost extremal size, that is, typically, $e(G)=ex(n,K_{r+1})-O(n)$. We show that such graphs must have a large amount of 'symmetry', in particular that all but very few vertices of $G$…

The classic extremal problem is that of computing the maximum number of edges in an $F$-free graph. In the case where $F=K_{r+1}$, the extremal number was determined by Tur\'an. Later results, known as supersaturation theorems, proved that…

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…

The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove…

Given a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-saturated if it is $\mathcal{F}$-free but the addition of any missing edge creates a copy of some $F \in \mathcal{F}$. The study of the minimum number of edges in…

A graph $\Gamma$ is said to be distance-balanced if for any edge $uv$ of $\Gamma$, the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$, and it is called nicely distance-balanced if…

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $\alpha_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with…

Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum cardinality of…

We study a Tur\'an-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a member from…

Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. For $k\ge 0$ an integer, a subset $I_k$ of $V(G)$ is called a $k$-nearly independent vertex subset of $G$ if $I_k$ induces a subgraph of size $k$ in $G$.…

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…

The binding number $b(G)$ of a graph, introduced by Woodall [J. Combin. Theory, Ser. B, 1973], is a central topic of both structural and extremal graph theory. It is closely related to fundamental combinatorial and structural properties of…

We say that two vertices are twins if they have the same neighbourhood and that a graph is $K_r$-saturated if it does not contain $K_r$ but adding any new edge to it creates a $K_r$. In 1964, Erd\H{o}s, Hajnal and Moon showed that…

We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.