A d-uniform hypergraph H is a sum hypergraph iff there is a finite S ⊆ IN such that H is isomorphic to the hypergraph H+d (S) = (V, E), where V = S and E = {{v1, . . . , vd} : (i 6= j ⇒ vi 6= vj)∧ ∑d i=1 vi ∈ S}. For an arbitrary d-uniform hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices w1, . . . , wσ 6∈ V such that H ∪ {w1, . . . , wσ} is a sum hyp...

The hypergraph of the Fano plane is the unique 3-uniform hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. This hypergraph is not 2-colorable, but becomes so on deleting any hyperedge from it. We show that taking uniformly at random a labeled 3-uniform hypergraph H on n vertices not containing the hypergraph of the Fano plane, H turns out t...

The Transversal Hypergraph Problem is the problem of computing, given a hypergraph, the set of its minimal transversals, i.e. the hypergraph whose hyperedges are all minimal hitting sets of the given one. This problem turns out to be central in various fields of Computer Science. We present and experimentally evaluate a heuristic algorithm for the problem, which seems able to handle large insta...

Let F be a strictly k-balanced k-uniform hypergraph with e(F ) ≥ |F | − k + 1 and maximum co-degree at least two. The random greedy F -free process constructs a maximal F -free hypergraph as follows. Consider a random ordering of the hyperedges of the complete k-uniform hypergraph K n on n vertices. Start with the empty hypergraph on n vertices. Successively consider the hyperedges e of K n in ...

We provide an exposition of the hypergraph models for parallel sparse matrix-vector multiplies based on one-dimensional (1D) matrix partitioning. Our aim is to emphasize the expressive power of the hypergraph models. We first set forth an elementary hypergraph model in which vertices represent the data elements of a matrix-vector multiply operation and nets encode data dependencies. We then app...

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n⌈k/3⌉) for any k ≥ 3, where n is the number of vertices. Moreover, we can do better when k ≡ 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n⌈k/3⌉−1/2). This result disproves a conjecture of Babai [1], which conje...

A hypergraph is color-critical if deleting any edge or vertex reduces the chromatic number; a color-critical hypergraph with chromatic number k is k-critical. Every k-chromatic hypergraph contains a k-critical hypergraph, so one can study chromatic number by studying the structure of k-critical (hyper)graphs. There is vast literature on k-critical graphs and hypergraphs. Many references can be ...

Graph and hypergraph partitioning are important problems with applications in several areas including parallel processing, VLSI design automation, physical mapping of chromosomes and data classification. The problem is to divide the vertices of a hypergraph into k similar–sized blocks such that a given cost function defined over the hypergraph is optimized. The intent of this paper is to expose...

A secret sharing scheme is a method to distribute a secret among a set of participants, such that only qualified subsets of the participants can recover the secret. A secret sharing scheme is perfect if any unqualified subset obtains no information regarding the master key. The collection of qualified subsets is called access structure. In a hypergraph, if the size of edges is equal to r, the h...