Generalizing the concept of dense hypergraph, we say that a hypergraph with n edges is weakly dense, if no k in half-open interval [2,n) degree more than k2 vertices. In our main result, prove famous Erdős–Faber–Lovász conjecture when dense.

Abstract Hypergraph computation has attracted much attention and shown apparent advantages in many application fields, such as computer vision, social networks, biomedicine. In this book, we systematically introduce the basic knowledge, algorithms, applications of hypergraph three parts discuss some recent progress direction.

One of the earliest results in Combinatorics is Mantel’s theorem from 1907 that the largest triangle-free graph on a given vertex set is complete bipartite. However, a seemingly similar question posed by Turán in 1941 is still open: what is the largest 3-uniform hypergraph on a given vertex set with no tetrahedron? This question can be considered a test case for the general hypergraph Turán pro...

We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. The research on this model for graphs has been started by Bohman et al. in [7], and continued in [8] and [16]. Here we obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergr...

Graph partitioning is an important and well studied problem in combinatorial scientific computing, and is commonly used to reduce communication in parallel computing. Different models (graph, hypergraph) and objectives (edge cut, boundary vertices) have been proposed. Hypergraph partitioning has become increasingly popular over the last decade. Its main strength is that it accurately captures c...

We consider hypergraphs with unordered hyperedges of unbounded rank. Each hypergraph has a unique modular decomposition, which is a tree, the nodes of which consist of certain prime subhypergraphs of the considered hypergraph. One can define this decomposition by Monadic Second-order (MS) logical formulas. Such a hypergraph is convex if the vertices are linearly ordered in such a way that the h...

Adaptive scientific computations require that periodic repartitioning (load balancing) occur dynamically to maintain load balance. Hypergraph partitioning is a successful model for minimizing communication volume in scientific computations, and partitioning software for the static case is widely available. In this paper, we present a new hypergraph model for the dynamic case, where we minimize ...

In this paper, we show that the eigenvectors of the zero Laplacian and signless Lapacian eigenvalues of a k-uniform hypergraph are closely related to some configured components of that hypergraph. We show that the components of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have the same modulus. Moreover, under a canonical regularization, the phases of the components of t...

In this paper we present a graph property with sensitivity Θ( √ n), where n = (v 2 ) is the number of variables, and generalize it to a k-uniform hypergraph property with sensitivity Θ( √ n), where n = (v k ) is again the number of variables. This yields the smallest sensitivity yet achieved for a k-uniform hypergraph property. We then show that, for even k, there is a k-uniform hypergraph prop...

The Transversal Hypergraph Generation is the problem of generating, given a hypergraph, the set of its minimal transversals, i.e., the hypergraph whose hyperedges are the minimal hitting sets of the given one. The purpose of this paper is to present an efficient and practical algorithm for solving this problem. We show that the proposed algorithm operates in a way that rules out regeneration an...