Related papers: An Improved Threshold for the Minimum Degree Krusk…

Let $C_6^3$ be the 3-uniform hypergraph on $\{1,\dots, 6\}$ with edges $123, 345,561$, which can be seen as the triangle in 3-uniform hypergraphs. For sufficiently large $n$ divisible by 6, we show that every $n$-vertex 3-uniform hypergraph…

The problem of determining the optimal minimum degree condition for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s} was solved by Zhao. Later Hladk\'y and Schacht, and Czygrinow and DeBiasio…

Many problems in extremal graph theory correspond to questions involving homomorphisms into a fixed image graph. Recently, there has been interest in maximizing the number of homomorphisms from graphs with a fixed number of vertices and…

Let $G$ be a graph and $v$ any vertex of $G$. We define the degenerate degree of $v$, denoted by $\zeta(v)$ as $\zeta(v)={\max}_{H: v\in H}~\delta(H)$, where the maximum is taken over all subgraphs of $G$ containing the vertex $v$. We show…

A family F is intersecting if any two members have a nonempty intersection. Erdos, Ko, and Rado showed that |F|\leq {n-1\choose k-1} holds for an intersecting family of k-subsets of [n]:={1,2,3,...,n}, n\geq 2k. For n> 2k the only extremal…

Let r, s >= 2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph K_n is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well…

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $G$ is $(k - 1)$-colorable, and if $L$ is a list-assignment for $G$, then $G$ is \textit{$L$-critical} if $G$ is not $L$-colorable but every proper induced…

Suppose $0 < p \le \infty$. For a simple graph $G$ with a vertex-degree sequence $d_1, \dots, d_n$ satisfying $(d_1^p + \dots + d_n^p)^{1/p} \le C$, we prove asymptotically sharp upper bounds on the number of $t$-cliques in $G$. This result…

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell\leq k-1$, we show that every $k$-graph on $n$…

Suppose $1\le \ell <k$ such that $(k-\ell)\nmid k$. Given an $n$-vertex $k$-uniform hypergraph $\mathcal H$, for all $k/2<\ell< 3k/4$ and sufficiently large $n\in (k-\ell)\mathbb N$, we prove that if $\mathcal H$ has minimum co-degree at…

Kostochka and Yancey resolved a famous conjecture of Ore on the asymptotic density of $k$-critical graphs by proving that every $k$-critical graph $G$ satisfies $|E(G)| \geq (\frac{k}{2} - \frac{1}{k-1})|V(G)| - \frac{k(k-3)}{2(k-1)}$. The…

The graph removal lemma is a fundamental result in extremal graph theory which says that for every fixed graph $H$ and $\varepsilon > 0$, if an $n$-vertex graph $G$ contains $\varepsilon n^2$ edge-disjoint copies of $H$ then $G$ contains…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is called minimal if for any edge $e\in…

An important question in extremal graph theory raised by Vera T. S\'os asks to determine for a given integer $t\ge 3$ and a given positive real number $\delta$ the asymptotically supremal edge density $f_t(\delta)$ that an $n$-vertex graph…

Given hypergraphs H and F, an F-factor in H is a spanning subgraph consisting of vertex disjoint copies of F. Let K_4^3-e denote the 3-uniform hypergraph on 4 vertices with 3 edges. We show that for \gamma>0 there exists an integer n_0 such…

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta(G)=2$. We show that in…

We study the $F$-decomposition threshold $\delta_F$ for a given graph $F$. Here an $F$-decomposition of a graph $G$ is a collection of edge-disjoint copies of $F$ in $G$ which together cover every edge of $G$. (Such an $F$-decomposition can…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of…

Problem of finding an optimal upper bound for the chromatic no. of a (3 Times K1)-free graph is still open and pretty hard. Here we prove that for a (3 Times K1)-free graph G with maximum degree greater than or equal to 8, {\chi} is less…

Given integer $k$ and a $k$-graph $F$, let $t_{k-1}(n,F)$ be the minimum integer $t$ such that every $k$-graph $H$ on $n$ vertices with codegree at least $t$ contains an $F$-factor. For integers $k\geq3$ and $0\leq\ell\leq k-1$, let…