In this note, we analyze two random greedy processes on sparse random graphs and hypergraphs with a given degree sequence. First we analyze the matching process, which builds a set of disjoint edges one edge at a time; then we analyze the independent process, which builds an independent set of vertices one vertex at a time. We use the differential equations method and apply a general theorem of Warnke. Our main contribution is to significantly reduce the associated systems of differential equations and simplify the expression for the final size of the matching or independent set.
We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.
We determine the unique hypergraphs with maximum spectral radius among all connected $k$-uniform ($k\geq 3$) unicyclic hypergraphs with matching number at least $z$, and among all connected $k$-uniform ($k\geq 3$) unicyclic hypergraphs with a given matching number, respectively.
We give a new proof of König's theorem and generalize the Gallai-Edmonds decomposition to balanced hypergraphs in two different ways. Based on our decompositions we give two new characterizations of balanced hypergraphs and show some properties of matchings and vertex cover in balanced hypergraphs.
We find the exact formula for the minimal number of edges of hypergraph which guaranteed fractional matching of cardinality $s$ in the case when $sn$ is integer.
A graph $G$ whose edges are coloured (not necessarily properly) contains a full rainbow matching if there is a matching $M$ that contains exactly one edge of each colour. We refute several conjectures on matchings in hypergraphs and full rainbow matchings in graphs, made by Aharoni and Berger and others.
A matching in a hypergraph $H$ is a set of pairwise vertex disjoint edges in $H$ and the matching number of $H$ is the maximum cardinality of a matching in $H$. A transversal in $H$ is a subset of vertices in $H$ that has a nonempty intersection with every edge of $H$. The transversal number $τ(H)$ of $H$ is the minimum cardinality of a transversal in $H$. A hypergraph $H$ is an intersecting hypergraph if every two distinct edges of $H$ have a non-empty intersection. Equivalently, $H$ is an intersecting hypergraph if and only if it has matching number one. In this paper we study the extremal b…
We prove a generalized Ramsey-type result on large 2-coloured matchings in a 3-coloured complete 3-uniform hypergraph, supporting a conjecture by A. Gyárfás.
In this note, we determine the maximum number of edges of a $k$-uniform hypergraph, $k\ge 3$, with a unique perfect matching. This settles a conjecture proposed by Snevily.
Extending the notion of (random) $k$-out graphs, we consider when the $k$-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each $r$ there is a $k=k(r)$ such that the $k$-out $r$-uniform hypergraph on $n$ vertices has a perfect fractional matching with high probability (i.e., with probability tending to $1$ as $n\to \infty$) and prove an analogous result for $r$-uniform $r$-partite hypergraphs. This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings. As a f…
We determine the \emph{exact} minimum $\ell$-degree threshold for perfect matchings in $k$-uniform hypergraphs when the corresponding threshold for perfect fractional matchings is significantly less than $\frac{1}{2} \binom{n}{k- \ell}$. This extends our previous results that determine the minimum $\ell$-degree thresholds for perfect matchings in $k$-uniform hypergraphs for all $\ell\ge k/2$ and provides two new (exact) thresholds: $(k,\ell)=(5,2)$ and $(7,3)$.
In 1965 Erdős conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative for k=3 and n is large enough.
A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is `pseudorandom', meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.
Let $H$ be a $3$-partite $3$-uniform hypergraph, i.e. a $3$-uniform hypergraph such that every edge intersects every partition class in exactly one vertex, with each partition class of size $n$. We determine a Dirac-type vertex degree threshold for perfect matchings in $3$-partite $3$-uniform hypergraphs.
The aim of the present paper is to prove that the maximum number of edges in a 3-uniform hypergraph on n vertices and matching number s is max{\binom(3s+2,3), \binom(n,3) - \binom(n-s,3)} for all n,s, n >= 3s+2.
A hypergraph $H$ is properly colored if for every vertex $v\in V(H)$, all the edges incident to $v$ have distinct colors. In this paper, we show that if $H_{1}$, \cdots, $H_{s}$ are properly-colored $k$-uniform hypergraphs on $n$ vertices, where $n\geq3k^{2}s$, and $e(H_{i})>{{n}\choose {k}}-{{n-s+1}\choose {k}}$, then there exists a rainbow matching of size $s$, containing one edge from each $H_i$. This generalizes some previous results on the Erdős Matching Conjecture.
Given positive integers k\geq 3 and r where k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, and builds on work in 'Exact minimum degree thresholds for perfect matchings in uniform hypergraphs', where we proved the result for k divisible by 4.
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 Erdős Matching Conjecture and convert the result to the minimum $d$-degree setting by an approach of Kühn, Osthus and Townsend. To obtain exact thresholds, we also apply a result of Treglown and Zhao.
Suppose $k\nmid n$ and $H$ is an $n$-vertex $k$-uniform hypergraph. A near perfect matching in $H$ is a matching of size $\lfloor n/k\rfloor$. We give a divisibility barrier construction that prevents the existence of near perfect matchings in $H$. This generalizes the divisibility barrier for perfect matchings. We give a conjecture on the minimum $d$-degree threshold forcing a (near) perfect matching in $H$ which generalizes a well-known conjecture on perfect matchings. We also verify our conjecture in various cases. Our proof makes use of the lattice-based absorbing method that the author us…
Our main result improves bounds of Markstrom and Rucinski on the minimum d-degree which forces a perfect matching in a k-uniform hypergraph on n vertices. We also extend bounds of Bollobas, Daykin and Erdos by asymptotically determining the minimum vertex degree which forces a matching of size t < n/2(k-1) in a k-uniform hypergraph on n vertices. Further asymptotically tight results on d-degrees which force large matchings are also obtained. Our approach is to prove fractional versions of the above results and then translate these into integer versions.
Let $k$ and $n$ be two integers, with $k\geq 3$, $n\equiv 0\pmod k$, and $n$ sufficiently large. We determine the $(k-1)$-degree threshold for the existence of a rainbow perfect matchings in $n$-vertex $k$-uniform hypergraph. This implies the result of Rödl, Ruciński, and Szemerédi on the $(k-1)$-degree threshold for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs. In our proof, we identify the extremal configurations of closeness, and consider whether or not the hypergraph is close to the extremal configuration. In addition, we also develop a novel absorbing device an…
For any $γ>0$, 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 $n/k+γn$. 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 Karpiński, Ruciński and Szymańska completely. Our proof uses a lattice-based absorbing method.
Let $\mathcal{U}(n,k)$ and $Γ(n,k)$ be the set of the $k$-uniform linear and nonlinear unicyclic hypergraphs having perfect matchings with $n$ vertices respectively, where $n\geq k(k-1)$ and $k\geq 3$. By using some techniques of transformations and constructing the incidence matrices for the hypergraphs considered, we get the hypergraphs with the maximal spectral radii among three kinds of hypergraphs, namely $\mathcal{U}(n,k)$ with $n= 2k(k-1)$ and $n\geq 9k(k-1)$, $Γ(n,k)$ with $n\geq k(k-1)$, and $\mathcal{U}(n,k)\cup Γ(n,k)$ with $n\geq 2k(k-1)$, where $k\geq 3$.
More than forty years ago, Erdős conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a basic instance of the hypergraph Turán problem (with a T-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all T < N/(3K^2). This improves upon the best previously known range T = O(N/K^3), which dates back to the 1970's.
Let $[n]$ denote the set $\{1, 2, \ldots, n\}$ and $\mathcal{F}^{(r)}_{n,k,a}$ be an $r$-uniform hypergraph on the vertex set $[n]$ with edge set consisting of all the $r$-element subsets of $[n]$ that contains at least $a$ vertices in $[ak+a-1]$. For $n\geq 2rk$, Frankl proved that $\mathcal{F}^{(r)}_{n,k,1}$ maximizes the number of edges in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$. Huang, Loh and Sudakov considered a multicolored version of the Erdős matching conjecture, and provided a sufficient condition on the number of edges for a multicolored hypergra…
Kühn, Osthus, and Treglown and, independently, Khan proved that if $H$ is a $3$-uniform hypergraph with $n$ vertices such that $n\in 3\mathbb{Z}$ and large, and $δ_1(H)>{n-1\choose 2}-{2n/3\choose 2}$, then $H$ contains a perfect matching. In this paper, we show that for $n\in 3\mathbb{Z}$ sufficiently large, if $F_1, \ldots, F_{n/3}$ are 3-uniform hypergrapghs with a common vertex set and $δ_1(F_i)>{n-1\choose 2}-{2n/3\choose 2}$ for $i\in [n/3]$, then $\{F_1,\dots, F_{n/3}\}$ admits a rainbow matching, i.e., a matching consisting of one edge from each $F_i$. This is done by converting the ra…
Given positive integers k and r where 4 divides k and k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, as well as the hypergraph removal lemma and a structural result of Keevash and Sudakov relating to the Turan number of the expanded triangle.
A matching in a hypergraph $\mathcal{H}$ is a set of pairwise disjoint hyperedges. The matching number $ν(\mathcal{H})$ of $\mathcal{H}$ is the size of a maximum matching in $\mathcal{H}$. A subset $D$ of vertices of $\mathcal{H}$ is a dominating set of $\mathcal{H}$ if for every $v\in V\setminus D$ there exists $u\in D$ such that $u$ and $v$ lie in an hyperedge of $\mathcal{H}$. The cardinality of a minimum dominating set of $\mathcal{H}$ is the domination number of $\mathcal{H}$, denoted by $γ(\mathcal{H})$. It was proved that $γ(\mathcal{H})\leq (r-1)ν(\mathcal{H})$ for $r$-uniform hypergra…
We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose that H is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than \binom{n-1}{2}-\binom{2n/3}{2}, then H contains a perfect matching. This bound is tight and answers a question of Han, Person and Schacht. More generally, we show that H contains a matching of size d\le n/3 if its minimum vertex degree is greater than \binom{n-1}{2}-\binom{n-d}{2}, which is also best possible. This extends a result of Bollob…
We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-graph without isolated vertex. More precisely, suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and divisible by $3$. If $H$ contains no isolated vertex and $deg(u)+ deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching. This bound is tight.
In this paper, we study degree conditions for the existence of large matchings in uniform hypergraphs. We prove that for integers $k,l,n$ with $k\ge 3$, $k/2<l<k$, and $n$ large, if $H$ is a $k$-uniform hypergraph on $n$ vertices and $δ_{l}(H)>{n-l\choose k-l}-{(n-l)-(\lceil n/k \rceil-2)\choose 2}$, then $H$ has a matching covering all but a constant number of vertices. When $l=k-2$ and $k\ge 5$, such a matching is near perfect and our bound on $δ_l(H)$ is best possible. When $k=3$, with the help of an absorbing lemma of Hán, Person, and Schacht, our proof also implies that $H$ has a perfect …
In this paper we prove an optimal co-degrees resilience property for the binomial $k$-uniform hypergraph model $H_{n,p}^k$ with respect to perfect matchings. That is, for a sufficiently large $n$ which is divisible by $k$, and $p\geq C_k\log_n/n$, we prove that with high probability every subgraph $H\subseteq H^k_{n,p}$ with minimum co-degree (meaning, the number of supersets every set of size $k-1$ is contained in) at least $(1/2+o(1))np$ contains a perfect matching.
We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as \emph{online sprinkling}. As an illustrative application of this method, we show that for any fixed integer $k\geq 3$, the binomial $k$-uniform random hypergraph $H^{k}_{n,p}$ contains $N:=(1-o(1))\binom{n-1}{k-1}p$ edge-disjoint perfect matchings, provided $p\geq \frac{\log^{C}n}{n^{k-1}}$, where $C:=C(k)$ is an integer depending only on $k$. Our result for $N$ is asymptotically best optimal and for $p$ is optimal up to the $polylog(n)$ factor.
Let $H$ be a $k$-uniform hypergraph on $n$ vertices where $n$ is a sufficiently large integer not divisible by $k$. We prove that if the minimum $(k-1)$-degree of $H$ is at least $\lfloor n/k \rfloor$, then $H$ contains a matching with $\lfloor n/k\rfloor$ edges. This confirms a conjecture of Rödl, Ruciński and Szemerédi, who proved that the minimum $(k-1)$-degree $n/k+O(\log n)$ suffices. More generally, we show that $H$ contains a matching of size $d$ if its minimum codegree is $d<n/k$, which is also best possible.
Let $n$ be a sufficiently large integer with $n\equiv 0\pmod 4$ and let $F_i \subseteq{[n]\choose 4}$ where $i\in [n/4]$. We show that if each vertex of $F_i$ is contained in more than ${n-1\choose 3}-{3n/4\choose 3}$ edges, then $\{F_1, \ldots ,F_{n/4}\}$ admits a rainbow matching, i.e., a set of $n/4$ edges consisting of one edge from each $F_i$. This generalizes a deep result of Khan on perfect matchings in 4-uniform hypergraphs.
Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $ν(G)$ denote the maximum size of a matching in $H$, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed $r, k, s \in \mathbb{N}$: $r$-Coloring in hypergraphs $G$ with edge size at most $k$ and $ν(G) \leq s$; $r$-Precoloring Extension in $k$-uniform hypergraphs $G$ with $ν(G) \leq s$; $r$-Precoloring Extension in hypergraphs $G$ with edge size at most $k$ and $ν(G) \le…
We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges. This result can be viewed as a special case of the degree version of a well-known conjecture of Erdős on hypergraph matchings. Improving the work of Bollobás, Daykin, and Erd\H os from 1976, we show that given integers $n, k, s$ with $n\ge 3k^2 s$, every $k$-uniform hypergraph $H$ on $n$ vertices with minimum vertex degree greater than $\binom{n-1}{k-1}-\binom…
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on…
Determine the size of $r$-graphs with given graph parameters is an interesting problem. Chvátal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014) studied the same problem for linear $3$-graphs with restricted matching number and maximum degree. In this paper, we give a tight upper bound of the size of $3$-graphs with bounded codegree and matching number.
A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost-regular, uniform hypergraph $\mathcal{H}$ with small maximum codegree has an almost-perfect matching. We extend this result by obtaining a ``conflict-free'' matching, where conflicts are encoded via a collection $\mathcal{C}$ of subsets $C\subseteq E(\mathcal{H})$. We say that a matching $\mathcal{M}\subseteq E(\mathcal{H})$ is conflict-free if $\mathcal{M}$ does not contain an element of $\mathcal{C}$ as a subset. Under natural assumptions on $\mathcal{C}$, we prove that $\mathcal{H}$ has a conflict-free, almost-p…
We give a new approach to handling hypergraph regularity. This approach allows for vertex-by-vertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.
Let $r \ge 2$ be a fixed constant and let $ {\mathcal H}$ be an $r$-uniform, $D$-regular hypergraph on $N$ vertices. Assume further that $ D \to \infty$ as $N \to \infty$ and that degrees of pairs of vertices in ${\mathcal H}$ are at most $L$ where $L \ = D/ (\log N)^{ω(1)}$. We consider the random greedy algorithm for forming a matching in $ \mathcal{H}$. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when…
Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erdős matching conjecture: For positive integers $n,k,m$ with $n\ge km$, if each of the families $F_1,\ldots, F_m\subseteq {[n]\choose k}$ has size more than $\max\{\binom{n}{k} - \binom{n-m+1}{k}, \binom{km-1}{k}\}$, then there exist pairwise disjoint subsets $e_1,\dots, e_m$ such that $e_i\in F_i$ for all $i\in [m]$. We prove that there exists an absolute constant $n_0$ such that this rainbow version holds for $k=3$ and $n\geq n_0$. We convert this rainbow matching problem to a matching …
Let $n, s$ be positive integers such that $n$ is sufficiently large and $s\le n/3$. Suppose $H$ is a 3-uniform hypergraph of order $n$. If $H$ contains no isolated vertex and $deg(u)+ deg(v) > 2(s-1)(n-1)$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a matching of size $s$. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when $s= n/3$.
Let $H$ be a $k$-graph on $n$ vertices, with minimum codegree at least $n/k + cn$ for some fixed $c > 0$. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in $H$ or a certificate that none exists. This essentially solves a problem of Karpiński, Ruciński and Szymańska; Szymańska previously showed that this problem is NP-hard for a minimum codegree of $n/k - cn$. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.
Let $σ$ be a partition of the positive integer $r$. A $σ$-hypergraph $H=H(n,r,q|σ)$ is an $r$-uniform hypergraph on $nq$ vertices which are partitioned into $n$ classes $V_1, V_2, \ldots, V_n$ each containing $q$ vertices. An $r$-subset $K$ of vertices is an edge of the hypergraph if the partition of $r$ formed by the non-zero cardinalities $|K\cap V_i|, 1\leq i \leq n,$ is $σ$. In earlier works we have considered colourings of the vertices of $H$ which are constrained such that any edge has at least $α$ and at most $β$ vertices of the same colour, and we have shown that interesting results ca…
Let $G$ be a simple graph that is properly edge coloured with $m$ colours and let $\M=\{M_1,\ldots, M_m\}$ be the set of $m$ matchings induced by the colours in $G$. Suppose that $m\le n-n^{c}$, where $c>9/10$, and every matching in $\M$ has size $n$. Then $G$ contains a full rainbow matching, i.e.\ a matching that contains exactly one edge from $M_i$ for each $1\le i\le m$. This answers an open problem of Pokrovskiy and gives an affirmative answer to a generalisation of a special case of a conjecture of Aharoni and Berger. Related results are also found for multigraphs with edges of bounded m…
Entropy Bounds for Perfect Matchings in Bipartite Hypergraphs
Georgia Institute of Technology（佐治亚理工学院）
本文研究了双图超图中完美匹配的数量上限，证明了在最大余度较小的均匀超图中，A-完美匹配的数量上限，并利用该结果证明了当n为奇数且n≡0 mod 3时，存在阶数为n的拉丁方，其至多有(n/e^{2.117})^n个transversals，同时展示了k-均匀D-正则超图的proper q-边着色数量上限。
