Perfect Matchings in Hypergraphs and the Erd\H{o}s matching conjecture

We prove a new upper bound for the minimum $d$-degree threshold for perfect matchings in $k$-uniform hypergraphs when $d<k/2$. As a consequence, this determines exact values of the threshold when $0.42k \le d < k/2$ or when $(k,d)=(12,5)$ or $(17,7)$. Our approach is to give an upper bound on the Erd\H{o}s Matching Conjecture and convert the result to the minimum $d$-degree setting by an approach of K\"uhn, Osthus and Townsend. To obtain exact thresholds, we also apply a result of Treglown and Zhao.Comment: 6 pages, referees' comments incorporated, to appear in SIDM

Decision problem for Perfect Matchings in Dense k-uniform Hypergraphs

For any $\gamma>0$, Keevash, Knox and Mycroft constructed a polynomial-time algorithm to determine the existence of perfect matchings in any $n$-vertex $k$-uniform hypergraph whose minimum codegree is at least $n/k+\gamma n$. We prove a structure theorem that enables us to determine the existence of a perfect matching for any $k$-uniform hypergraph with minimum codegree at least $n/k$. This solves a problem of Karpi\'nski, Ruci\'nski and Szyma\'nska completely. Our proof uses a lattice-based absorbing method.Comment: Accepted by Transactions of the AMS. arXiv admin note: substantial text overlap with arXiv:1307.2608 by other author

On Perfect Matchings and tilings in uniform Hypergraphs

In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for $k\ge 3$, if $H$ is a $k$-graph on $n\in k\mathbb N$ vertices with independence number at most $n/p$ and minimum codegree at least $(1/p+o(1))n$, where $p$ is the smallest prime factor of $k$, then $H$ contains a perfect matching. Second, we show that if $H$ is a $3$-graph on $n\in 3\mathbb N$ vertices which does not contain any induced copy of $K_4^-$ (the unique $3$-graph with $4$ vertices and $3$ edges) and has minimum codegree at least $(1/3+o(1)))n$, then $H$ contains a perfect matching. Moreover, if we allow the matching to miss at most $3$ vertices, then the minimum degree condition can be reduced to $(1/6+o(1)))n$. Third, we show that if $H$ is a $3$-graph on $n\in 4\mathbb N$ vertices which does not contain any induced copy of $K_4^-$ and has minimum codegree at least $(1/8+o(1)))n$, then $H$ contains a perfect $Y$-tiling, where $Y$ represents the unique $3$-graph with $4$ vertices and $2$ edges. We also provide the examples showing that our minimum codegree conditions are asymptotically best possible. Our main tool for finding the perfect matching is a characterization theorem that characterizes the $k$-graphs with minimum codegree at least $n/k$ which contain a perfect matching.Comment: 11 pages, to appear in SIDM

Minimum codegree threshold for Hamilton l-cycles in k-uniform hypergraphs

For $1\le \ell<k/2$, we show that for sufficiently large $n$, every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\frac n{2 (k-\ell)}$ contains a Hamilton $\ell$-cycle. This codegree condition is best possible and improves on work of H\`an and Schacht who proved an asymptotic result.Comment: 22 pages, 0 figure. Accepted for publication in JCTA. arXiv admin note: text overlap with arXiv:1307.369

The maximum size of a non-trivial intersecting uniform family that is not a subfamily of the Hilton--Milner family

The celebrated Erd\H{o}s-Ko-Rado theorem determines the maximum size of a $k$-uniform intersecting family. The Hilton-Milner theorem determines the maximum size of a $k$-uniform intersecting family that is not a subfamily of the so-called Erd\H{o}s-Ko-Rado family. In turn, it is natural to ask what the maximum size of an intersecting $k$-uniform family that is neither a subfamily of the Erd\H{o}s-Ko-Rado family nor of the Hilton-Milner family is. For $k\ge 4$, this was solved (implicitly) in the same paper by Hilton-Milner in 1967. We give a different and simpler proof, based on the shifting method, which allows us to solve all cases $k\ge 3$ and characterize all extremal families achieving the extremal value.Comment: 15 pages, 1 figure; To appear in Proc. Amer. Math. So

Forbidding Hamilton cycles in uniform hypergraphs

For $1\le d\le \ell< k$, we give a new lower bound for the minimum $d$-degree threshold that guarantees a Hamilton $\ell$-cycle in $k$-uniform hypergraphs. When $k\ge 4$ and $d< \ell=k-1$, this bound is larger than the conjectured minimum $d$-degree threshold for perfect matchings and thus disproves a well-known conjecture of R\"odl and Ruci\'nski. Our (simple) construction generalizes a construction of Katona and Kierstead and the space barrier for Hamilton cycles.Comment: 6 pages, 0 figur

On hypergraphs without loose cycles

Recently, Mubayi and Wang showed that for $r\ge 4$ and $\ell \ge 3$, the number of $n$-vertex $r$-graphs that do not contain any loose cycle of length $\ell$ is at most $2^{O( n^{r-1} (\log n)^{(r-3)/(r-2)})}$. We improve this bound to $2^{O( n^{r-1} \log \log n) }$.Comment: 6 page

Minimum vertex degree threshold for C43C_4^3-tiling

We prove that the vertex degree threshold for tiling \C_4^3 (the 3-uniform hypergraph with four vertices and two triples) in a 3-uniform hypergraph on $n\in 4\mathbb N$ vertices is $\binom{n-1}2 - \binom{\frac34 n}2+\frac38n+c$, where $c=1$ if $n\in 8\mathbb N$ and $c=-\frac12$ otherwise. This result is best possible, and is one of the first results on vertex degree conditions for hypergraph tiling.Comment: 16 pages, 0 figure. arXiv admin note: text overlap with arXiv:0903.2867 by other author

On multipartite Hajnal-Szemer\'edi theorems

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 $n$ is sufficiently large, then $G$ contains a $K_3$-factor (a spanning subgraph consisting of $n$ vertex-disjoint copies of $K_3$) except that $G$ is one particular graph. Martin and Szemer\'edi \cite{MaSz} proved that $G$ contains a $K_4$-factor when $\delta^*(G)\ge 3/4n$ and $n$ is sufficiently large. Both results were proved by the Regularity Lemma. In this paper we give a proof of these two results by the absorbing method. Our absorbing lemma actually works for all $k\ge 3$.Comment: 15 pages, no figur

Minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs

We show that for sufficiently large $n$, every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $\binom{n-1}2 - \binom{\lfloor\frac34 n\rfloor}2 + c$, where $c=2$ if $n\in 4\mathbb{N}$ and $c=1$ if $n\in 2\mathbb{N}\setminus 4\mathbb{N}$, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Bu\ss, H\`an and Schacht who proved the corresponding asymptotical result.Comment: 23 pages, 1 figure, Accepted for publication in JCT