Perfect matchings in random s-uniform hypergraphs

Let E = {X₁, X₂…, X_m} where the X_i ? V for 1 ≤ i ≤ m are distinct. The hypergraph G = (V, E) is said to be s-uniform if |X₁| = s for 1 ≤ i ≤ m. A set of edges M = {X_i : i ? I } is a perfect matching if (i) i ≠ j ? I implies X_i ∩ X_i = 0, and (ii) ∪_i?I X_i = V. In this article we consider the question of whether a random s-uniform hypergraph contains a perfect matching. Let s ≥ 3 be fixed and m/n^4/3 → ∞. We show that an s-uniform hypergraph with m edges chosen uniformly from [74] contains a perfect matching with high probability. This improves an earlier result of Schmidt and Shamir who showed that m/n^3/2 → ∞ suffices. © 1995 John Wiley & Sons, Inc.     

Perfect matchings in random uniform hypergraphs

Let ??_k(n, p) be the random k‐uniform hypergraph on V = [n] with edge probability p. Motivated by a theorem of Erd?s and Rényi 7 regarding when a random graph G(n, p) = ??₂(n, p) has a perfect matching, the following conjecture may be raised. (See J. Schmidt and E. Shamir 16 for a weaker version.) Conjecture. Let k|n for fixed k ≥ 3, and the expected degree d(n, p) = p(). Then (Erd?s and Rényi 7 proved this for G(n, p).) Assuming d(n, p)/n^1/2 → ∞, Schmidt and Shamir 16 were able to prove that ??_k(n, p) contains a perfect matching with probability 1 ? o(1). Frieze and Janson 8 showed that a weaker condition d(n, p)/n^1/3 → ∞ was enough. In this paper, we further weaken the condition to A condition for a similar problem about a perfect triangle packing of G(n, p) is also obtained. A perfect triangle packing of a graph is a collection of vertex disjoint triangles whose union is the entire vertex set. Improving a condition p ≥ cn^?2/3+1/15 of Krivelevich 12 , it is shown that if 3|n and p ? n^?2/3+1/18, then © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 111–132, 2003     

Perfect matchings in balanced hypergraphs

We generalize Hall's condition for the existence of a perfect matching in a bipartite graph, to balanced hypergraphs.This work was partially supported in part by NSF grants DMI-9424348, DMS-9509581 and ONR grant N00014-89-J-1063. Ajai Kapoor was also supported by a grant from Gruppo Nazionale Delle Riccerche-CNR. Finally, we acknowledge the support of Laboratiore ARTEMIS, Université Joseph Fourier, Grenoble.     

Packing perfect matchings in random hypergraphs

We introduce a new procedure for generating the binomial random graph/hypergraph models, referred to as online sprinkling. As an illustrative application of this method, we show that for any fixed integer , the binomial ‐uniform random hypergraph contains edge‐disjoint perfect matchings, provided , where is an integer depending only on . Our result for is asymptotically optimal and for is optimal up to the factor. This significantly improves a result of Frieze and Krivelevich.     

Almost perfect matchings in random uniform hypergraphs

We consider the following model H_r(n, p) of random r-uniform hypergraphs. The vertex set consists of two disjoint subsets V of size | V | = n and U of size | U | = (r − 1)n. Each r-subset of V × (_r−1^U) is chosen to be an edge of H ε H_r(n, p) with probability p = p(n), all choices being independent. It is shown that for every 0 < < 1 if P = (C ln n)/n^r−1 with C = C() sufficiently large, then almost surely every subset V₁ V of size | V₁ | = (1 − )n is matchable, that is, there exists a matching M in H such that every vertex of V₁ is contained in some edge of M.     

Maximum degree and fractional matchings in uniform hypergraphs

Let ℋ be a family ofr-subsets of a finite setX. SetD(ℋ)= |{E:x∈E∈ℋ}|, (maximum degree). We say that ℋ is intersecting if for anyH,H′ ∈ ℋ we haveH ∩H′ ≠ 0. In this case, obviously,D(ℋ)≧|ℋ|/r. According to a well-known conjectureD(ℋ)≧|ℋ|/(r−1+1/r). We prove a slightly stronger result. Let ℋ be anr-uniform, intersecting hypergraph. Then either it is a projective plane of orderr−1, consequentlyD(ℋ)=|ℋ|/(r−1+1/r), orD(ℋ)≧|ℋ|/(r−1). This is a corollary to a more general theorem on not necessarily intersecting hypergraphs.     

Perfect matchings in uniform hypergraphs with large minimum degree

A perfect matching in a k-uniform hypergraph on n vertices, n divisible by k, is a set of n/k disjoint edges. In this paper we give a sufficient condition for the existence of a perfect matching in terms of a variant of the minimum degree. We prove that for every k≥3 and sufficiently large n, a perfect matching exists in every n-vertex k-uniform hypergraph in which each set of k−1 vertices is contained in n/2+Ω(logn) edges. Owing to a construction in [D. Kühn, D. Osthus, Matchings in hypergraphs of large minimum degree, J. Graph Theory 51 (1) (2006) 269–280], this is nearly optimal. For almost perfect and fractional perfect matchings we show that analogous thresholds are close to n/k rather than n/2.     

Perfect matchings in random intersection graphs

Let W ₁,??,W _n be independent random subsets of [m]={1,??,m}. Assuming that each W _i is uniformly distributed in the class of d-subsets of?[m] we study the uniform random intersection graph G _s (n,m,d) on the vertex set {W ₁,??W _n }, defined by the adjacency relation: W _i ??W _j whenever |W _i ??W _j |?Rs. For even?n we show that as n,m???? the edge density threshold for the property that G _s (n,m,d) contains a perfect matching is asymptotically the same as that for G _s (n,m,d) being connected.     

Perfect matchings in large uniform hypergraphs with large minimum collective degree

We define a perfect matching in a k-uniform hypergraph H on n vertices as a set of ⌊n/k⌋ disjoint edges. Let δk−1(H) be the largest integer d such that every (k−1)-element set of vertices of H belongs to at least d edges of H.In this paper we study the relation between δk−1(H) and the presence of a perfect matching in H for k?3. Let t(k,n) be the smallest integer t such that every k-uniform hypergraph on n vertices and with δk−1(H)?t contains a perfect matching.For large n divisible by k, we completely determine the values of t(k,n), which turn out to be very close to n/2−k. For example, if k is odd and n is large and even, then t(k,n)=n/2−k+2. In contrast, for n not divisible by k, we show that t(k,n)∼n/k.In the proofs we employ a newly developed “absorbing” technique, which has a potential to be applicable in a more general context of establishing existence of spanning subgraphs of graphs and hypergraphs.     

Optimal fractional matchings and covers in infinite hypergraphs: Existence and duality

We study fractional matchings and covers in infinite hypergraphs, paying particular attention to the following questions: Do fractional matchings (resp. covers) of maximal (resp. minimal) size exist? Is there equality between the supremum of the sizes of fractional matchings and the infimum of the sizes of fractional covers? (This is called weak duality.) Are there a fractional matching and a fractional cover that satisfy the complementary slackness conditions of linear programming? (This is called strong duality.) In general, the answers to all these questions are negative, but for certain classes of infinite hypergraphs (classified according to edge cardinalities and vertex degrees) we obtain positive results. We also consider the question of the existence of optimal fractional matchings and covers that assume rational values.     

Polynomial-time perfect matchings in dense hypergraphs

Let H be a k-graph on n   vertices, with minimum codegree at least n/k+cn$n/k+cn$ for some fixed c>0$c>0$. 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$n/k-cn$. 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.     

Nearly perfect matchings in regular simple hypergraphs

For every fixedk≥3 there exists a constantc _k with the following property. LetH be ak-uniform,D-regular hypergraph onN vertices, in which no two edges contain more than one common vertex. Ifk>3 thenH contains a matching covering all vertices but at mostc _k ND ^−1/(k−1). Ifk=3, thenH contains a matching covering all vertices but at mostc ₃ ND ^−1/2ln^3/2 D. This improves previous estimates and implies, for example, that any Steiner Triple System onN vertices contains a matching covering all vertices but at mostO(N ^1/2ln^3/2 N), improving results by various authors. Research supported in part by a USA-Israel BSF grant. Research supported in part by a USA-Israel BSF Grant.     

Fractional matchings and covers in infinite hypergraphs

A strong version of the duality theorem of linear programming is proved for fractional covers and matchings in countable graphs. It is conjectured to hold for general hypergraphs. In Section 2 we show that in countable hypergraphs there does not necessarily exist a maximal matchable set, contrary to the situation in graphs.     

Perfect matchings in random bipartite graphs with minimal degree at least 2

We show that a random bipartite graph with n+n vertices and cn random edges, and minimum degree at least two, has a perfect matching whp . © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 26, 2005     

Perfect matchings in hexagonal systems

A simple algorithm is developed which allows to decide whether or not a given hexagonal system has a perfect matching (and to find such a matching). This decision is also of chemical relevance since a hexagonal system is the skeleton of a benzenoid hydrocarbon molecule if and only if it has a perfect matching. Dedicated to Paul Erdős on his seventieth birthday     

Perfect matchings in hexagonal systems

This paper deals with perfect matchings in hexagonal systems. Counterexamples are given to Sachs's conjecture in this field. A necessary and sufficient condition for a hexagonal system to have a perfect matching is obtained.