Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

The order of the giant component of random hypergraphs

We establish central and local limit theorems for the number of vertices in the largest component of a random d‐uniform hypergraph H_d(n,p) with edge probability p = c/ , where c > (d ‐ 1)^‐1 is a constant. The proof relies on a new, purely probabilistic approach. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

List coloring triangle‐free hypergraphs

A triangle in a hypergraph is a collection of distinct vertices u, v, w and distinct edges e, f, g with , and . Johansson [Tech. report (1996)] proved that every triangle‐free graph with maximum degree Δ has list chromatic number . Frieze and Mubayi (Electron J Comb 15 (2008), 27) proved that every linear (meaning that every two edges share at most one vertex) triangle‐free triple system with maximum degree Δ has chromatic number . The restriction to linear triple systems was crucial to their proof. We provide a common generalization of both these results for rank 3 hypergraphs (edges have size 2 or 3). Our result removes the linear restriction from 8 , while reducing to the (best possible) result [Johansson, Tech. report (1996)] for graphs. In addition, our result provides a positive answer to a restricted version of a question of Ajtai Erd?s, Komlós, and Szemerédi (combinatorica 1 (1981), 313–317) concerning sparse 3‐uniform hypergraphs. As an application, we prove that if is the collection of 3‐uniform triangles, then the Ramsey number satisfies for some positive constants a and b. The upper bound makes progress towards the recent conjecture of Kostochka, Mubayi, and Verstraëte (J Comb Theory Ser A 120 (2013), 1491–1507) that where C₃ is the linear triangle. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 487–519, 2015     

Quasi-random hypergraphs

We introduce an equivalence class of varied properties for hypergraphs. Any hypergraph possessing any one of these properties must of necessity possess them all. Since almost all random hypergraphs share these properties, we term these properties quasi-random. With these results, it becomes quite easy to show that many natural explicit constructions result in hypergraphs which imitate random hypergraphs in a variety of ways.     

Tight Hamilton cycles in random uniform hypergraphs

In this paper we show that e/n is the sharp threshold for the existence of tight Hamilton cycles in random k ‐uniform hypergraphs, for all k ≥ 4. When k = 3 we show that 1/n is an asymptotic threshold. We also determine thresholds for the existence of other types of Hamilton cycles. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

The chromatic numbers of random hypergraphs

For a pair of integers 1≤γ<r, the γ-chromatic number of an r-uniform hypergraph H=(V, E) is the minimal k, for which there exists a partition of V into subsets T₁,…,T_k such that |e∩T_i|≤γ for every e∈E. In this paper we determine the asymptotic behavior of the γ-chromatic number of the random r-uniform hypergraph H_r(n, p) for all possible values of γ and for all values of p down to p=Θ(n^−r+1). © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 381–403, 1998     

Partitions of hypergraphs under variable degeneracy constraints

The paper deals with partitions of hypergraphs into induced subhypergraphs satisfying constraints on their degeneracy. Our hypergraphs may have multiple edges, but no loops. Given a hypergraph $H$ and a sequence $f=\left(f_1,f_2,\text{…},f_p\right)$ of $p\ge 1$ vertex functions $f_i:V\left(H\right)\to {\mathbb{N}}_0$ such that $f_1\left(v\right)+f_2\left(v\right)+\mathrm{?}+f_p\left(v\right)\ge d_H\left(v\right)$ for all $v\in V\left(H\right)$, we want to find a sequence $\left(H_1,H_2,\text{…},H_p\right)$ of vertex disjoint induced subhypergraphs containing all vertices of $H$ such that each hypergraph $H_i$ is strictly $f_i$‐degenerate, that is, for every nonempty subhypergraph $H\prime ?H_i$ there is a vertex $v\in V\left(H\prime \right)$ such that $d_{H\prime }\left(v\right)<f_i\left(v\right)$. Our main result in this paper says that such a sequence of hypergraphs exists if and only if $\left(H,f\right)$ is not a so‐called hard pair. Hard pairs form a recursively defined family of configurations, obtained from three basic types of configurations by the operation of merging a vertex. Our main result has several interesting applications related to generalized hypergraph coloring problems.     

A note on near‐optimal coloring of shift hypergraphs

As shown in the original work on the Lovász Local Lemma due to Erd?s & Lovász (Infinite and Finite Sets, 1975), a basic application of the Local Lemma answers an infinitary coloring question of Strauss, showing that given any integer set S, the integers may be k‐colored so that S and all its translates meet every color. The quantitative bounds here were improved by Alon, Kriz & Nesetril (Studia Scientiarum Mathematicarum Hungarica, 1995). We obtain an asymptotically optimal bound in this note, using the technique of iteratively applying the Lovász Local Lemma in order to prune dependencies. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 53–56, 2016     

Powers of tight Hamilton cycles in randomly perturbed hypergraphs

For k ≥ 2 and r ≥ 1 such that k + r ≥ 4, we prove that, for any α > 0, there exists ε > 0 such that the union of an n‐vertex k‐graph with minimum codegree and a binomial random k‐graph with on the same vertex set contains the rth power of a tight Hamilton cycle with high probability. This result for r = 1 was first proved by McDowell and Mycroft.     

Exploring hypergraphs with martingales

Recently, in [Random Struct Algorithm 41 (2012), 441–450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 (2007), 133–142], in turn based on ideas of Martin‐Löf [J Appl Probab 23 (1986), 265–282], Karp [Random Struct Alg 1 (1990), 73–93] and Aldous [Ann Probab 25 (1997), 812–854], to prove asymptotic normality of the number L₁ of vertices in the largest component of the random r‐uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L₁, and joint asymptotic normality of L₁ and the number M₁ of edges of in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 (2016), 21–75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325–352, 2017     

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.     

Discrepancy of random graphs and hypergraphs

Answering in a strong form a question posed by Bollobás and Scott, in this paper we determine the discrepancy between two random k‐uniform hypergraphs, up to a constant factor depending solely on k. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 47, 147–162, 2015     

Packing tight Hamilton cycles in 3‐uniform hypergraphs

Let H be a 3‐uniform hypergraph with n vertices. A tight Hamilton cycle C ? H is a collection of n edges for which there is an ordering of the vertices v₁,…,v_n such that every triple of consecutive vertices {v_i,v_i+1,v_i+2} is an edge of C (indices are considered modulo n ). We develop new techniques which enable us to prove that under certain natural pseudo‐random conditions, almost all edges of H can be covered by edge‐disjoint tight Hamilton cycles, for n divisible by 4. Consequently, we derive the corollary that random 3‐uniform hypergraphs can be almost completely packed with tight Hamilton cycles whp, for n divisible by 4 and p not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudo‐random digraphs with even numbers of vertices. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Tight Hamilton cycles in random hypergraphs

We give an algorithmic proof for the existence of tight Hamilton cycles in a random r‐uniform hypergraph with edge probability for every . This partly answers a question of Dudek and Frieze (Random Struct Algor 42 (2013), 374–385), who used a second moment method to show that tight Hamilton cycles exist even for where arbitrary slowly, and for . The method we develop for proving our result applies to related problems as well. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 446–465, 2015     

On the connectivity of proper colorings of random graphs and hypergraphs

Let Ω_q=Ω_q(H) denote the set of proper [q]‐colorings of the hypergraph H. Let Γ_q be the graph with vertex set Ω_q where two colorings σ,τ are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_n,m;k, k ≥ 2, the random k‐uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Γ_q is connected if d is sufficiently large and . This is optimal up to the first order in d. Furthermore, with a few more colors, we find that the diameter of Γ_q is O(n) w.h.p., where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber dynamics Markov Chain on Ω_q is ergodic w.h.p.     

On the maximum size of connected hypergraphs without a path of given length

In this note we asymptotically determine the maximum number of hyperedges possible in an $r$-uniform, connected $n$-vertex hypergraph without a Berge path of length $k$, as $n$ and $k$ tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity.