Related papers: Towards a weighted version of the Hajnal-Szemer\'e…

A classical result by Hajnal and Szemer\'edi from 1970 determines the minimal degree conditions necessary to guarantee for a graph to contain a $K_r$-factor. Namely, any graph on $n$ vertices, with minimum degree $\delta(G) \ge…

A seminal result of Hajnal and Szemer\'{e}di states that if a graph $G$ with $n$ vertices has minimum degree $\delta(G) \ge (r-1)n/r$ for some integer $r \ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples…

A $K_r$-factor of a graph $G$ is a collection of vertex-disjoint $r$-cliques covering $V(G)$. We prove the following algorithmic version of the classical Hajnal--Szemer\'edi Theorem in graph theory, when $r$ is considered as a constant.…

The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-packing. Fischer's conjecture states that the analogous result holds for all multipartite graphs except for…

A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemer\'edi states that if $G$ is a graph of order $n$ with minimum degree $\delta(G) \ge…

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the clique $K_r$ in $G$ covering every vertex of $G$. The famous Hajnal--Szemer\'edi theorem determines the minimum degree threshold for forcing a perfect…

We say that a graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemer\'edi theorem characterises the minimum degree that ensures a graph $G$…

We say that a (di)graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. The seminal Hajnal--Szemer\'edi theorem characterises the minimum degree that ensures a graph…

In this note we show the following strengthening of a multipartite version of the Hajnal--Szemer\'edi theorem. For an integer $r \ge 3$ and $\gamma > 0$, there exists a constant $C$ such that if $p\ge Cn^{-2/r}(\log n)^{1/{r \choose 2}}$…

Consider a host hypergraph $G$ which contains a spanning structure due to minimum degree considerations. We collect three results proving that if the edges of $G$ are sampled at the appropriate rate then the spanning structure still appears…

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 is minimal if for every edge, the deletion of…

Let K_r^- denote the graph obtained from K_r by deleting one edge. We show that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that every graph G whose order n\ge n_0 is divisible by r and whose minimum degree is at least…

Hajnal and Szemeredi proved that every graph G with |G|=ks and minimum degree at least k(s-1) contains k vertex disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every…

A graph of order $n$ is said to be $k$-\emph{factor-critical} $(0\le k<n)$ if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical…

The celebrated Corr\'{a}di--Hajnal Theorem~\cite{CH63} and the Hajnal--Szemer\'{e}di Theorem~\cite{HS70} determined the exact minimum degree thresholds for a graph on $n$ vertices to contain $k$ vertex-disjoint copies of $K_r$, for $r=3$…

We say that a graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect H-packings: Given positive integers n, r, D, we…

Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$…

Let $q$ be a positve integer, and $G$ be a $q$-partite simple graph on $qn$ vertices, with $n$ vertices in each vertex class. Let $\delta={k_q \over k_q+1}$, where $k_q=q+O(\log{q})$. If each vertex of $G$ is adjacent to at least $\delta n$…

Let $G$ be a $k$-partite graph with $n$ vertices in parts such that each vertex is adjacent to at least $\delta^*(G)$ vertices in each of the other parts. Magyar and Martin \cite{MaMa} proved that for $k=3$, if $\delta^*(G)\ge 2/3n $ and…

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $\delta(G) \ge \max \{ \frac{1}{2},\frac{r…