In this paper, the concept of hypergraph is extended to the fuzzy hypergraph. In the fuzzy hypergraph, the concepts of a-cut hypergraph, strength of edge and dual fuzzy hypergraph are developed. It is shown that the fuzzy hypergraph and a-cut hypergraph are useful to represent a fuzzy partition. An application example also shows that the strength of edge can be used to decompose the data set in...

Abstract. A k -hypertournament is a complete k -hypergraph with each k -edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a k hypertournament, the score si (losing score ri ) of a vertex vi is the number of edges containing vi in which vi is not the last element (in which vi is the last element). In this paper we obtain inequalities involv...

A cover of a hypergraph is a collection of edges whose union contains all vertices. Let H = (V, E) be a k-uniform, D-regular hypergraph on n vertices, in which no two vertices are contained in more than o(D/e log D) edges as D tends to infinity. Our results include that if k = o(log D), then there is a cover of (1 + o(1))n/k edges, extending the known result that this holds for fixed k. On the ...

In this thesis, we survey the current research into self-complementary hypergraphs, and present several new results. We characterize the cycle type of the permutations on n elements with order equal to a power of 2 which are k-complementing. The k-complementing permutations map the edges of a k-uniform hypergraph to the edges of its complement. This yields a test to determine whether a finite p...

Let G be a t uniform s regular linear hypergraph with r vertices It is shown that the number of independent sets I G in G satis es log I G r t O log ts s This leads to an improvement of a previous bound by Alon obtained for t i e for regular ordinary graphs It is also shown that for the Hamming graph H n q with vertices consisting of all n tuples over an alphabet of size q and edges connecting ...

We introduce a new variant of Szemerédi’s regularity lemma which we call the sparse regular approximation lemma (SRAL). The input to this lemma is a graph G of edge density p and parameters , δ, where we think of δ as a constant. The goal is to construct an -regular partition of G while having the freedom to add/remove up to δ|E(G)| edges. As we show here, this weaker variant of the regularity ...

A vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f(r,∆) be the smallest integer k such that each r-uniform hypergraph of maximum vertex degree ∆ has a conflict-free coloring with at most k colors. As shown by Tardos and Pach, similarly to a classical Brooks’ type theorem for hypergraphs, f(r,∆) ≤...

In this paper, we introduce a new problem called Tree-Residue Vertex-Breaking (TRVB): given a multigraph G some of whose vertices are marked “breakable,” is it possible to convert G into a tree via a sequence of “vertex-breaking” operations (replacing a degree-k breakable vertex by k degree-1 vertices, disconnecting the k incident edges)? We characterize the computational complexity of TRVB wit...

The analysis of several algorithms and data structures can be framed as a peeling process on a random hypergraph: vertices with degree less than k and their adjacent edges are removed until no vertices of degree less than k are left. Often the question is whether the remaining hypergraph, the k-core, is empty or not. In some settings, it may be possible to remove either vertices or edges from t...