Related papers: Codegree conditions for tilling balanced complete …

Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an $F$-covering if every vertex in $G$ is contained in a copy of $F$. For $1\le i \le k-1$, let $c_i(n,F)$ be the least integer such that every $n$-vertex $k$-uniform…

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an $n$-vertex graph $G$ with sublinear independence number. In this setting, we show that if $\delta(G) \ge n/3 + o(n)$ then…

For a fixed integer h>=1, let G be a tripartite graph with N vertices in each vertex class, N divisible by 6h, such that every vertex is adjacent to at least 2N/3+h-1 vertices in each of the other classes. We show that if N is sufficiently…

Given two $r$-uniform hypergraphs $F$ and $H$, we say that $H$ has an $F$-covering if every vertex in $H$ is contained in a copy of $F$. Let $c_{i}(n,F)$ be the least integer such that every $n$-vertex $r$-graph $H$ with…

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the graph $K_r$ in $G$ that covers all vertices of $G$. In this paper, we prove that the threshold for the existence of a perfect $K_{r}$-tiling of a…

We give, for each $k \geq 3$, the precise best possible minimum positive codegree condition for a perfect matching in a large $k$-uniform hypergraph $H$ on $n$ vertices. Specifically we show that, if $n$ is sufficiently large and divisible…

We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs $H$ with components of sublinear order. As a corollary, we recover and extend the work of K\"uhn and…

A recent paper of Balogh, Li and Treglown initiated the study of Dirac-type problems for ordered graphs. In this paper we prove a number of results in this area. In particular, we determine asymptotically the minimum degree threshold for…

In 1975 Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $\delta(G)\geq n+2n^{\frac{5}{6}}$…

For any bipartite graph $H$, we determine a minimum degree threshold for a balanced bipartite graph $G$ to contain a perfect $H$-tiling. We show that this threshold is best possible up to a constant depending only on $H$. Additionally, we…

For any $k\ge 3$ and $\ell \in [k-1]$ such that $(k,\ell) \ne (3,1)$, we show that any sufficiently large $k$-graph $G$ must contain a Hamilton $\ell$-cycle provided that it has no isolated vertices and every set of $k-1$ vertices contained…

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…

In the perfect tiling problem, we aim to cover the vertices of a hypergraph~$G$ with pairwise vertex-disjoint copies of a hypergraph $F$. There are three essentially necessary conditions for such a perfect tiling, which correspond to…

A fundamental theorem of Wilson states that, for every graph $F$, every sufficiently large $F$-divisible clique has an $F$-decomposition. Here a graph $G$ is $F$-divisible if $e(F)$ divides $e(G)$ and the greatest common divisor of the…

Given integers $ n \ge k >l \ge 1 $ and a $k$-graph $F$ with $|V(F)|$ divisible by $n$, define $t_l^k(n,F)$ to be the smallest integer $d$ such that every $k$-graph $H$ of order $n$ with minimum $l$-degree $\delta_l(H) \ge d $ contains an…

Let $G$ and $H$ be $k$-graphs ($k$-uniform hypergraphs); then a perfect $H$-packing in $G$ is a collection of vertex-disjoint copies of $H$ in $G$ which together cover every vertex of $G$. For any fixed $H$ let $\delta(H, n)$ be the minimum…

For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $r\in \mathbb{N}$, the $r$-independence number of $G$, denoted $\alpha_r(G)$ is the largest size of a…

A graph $H$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to…

Given two $k$-graphs $H$ and $F$, a perfect $F$-packing in $H$ is a collection of vertex-disjoint copies of $F$ in $H$ which together cover all the vertices in $H$. In the case when $F$ is a single edge, a perfect $F$-packing is simply a…

A graph $G$ admits an $H$-tiling if it contains a collection of vertex-disjoint copies of $H$. In this paper, we confirm a conjecture proposed by K\"{u}hn, Osthus, and Treglown by showing that for any given graph $H$, there exists a…