An r-uniform hypergraph is k-edge-hamiltonian iff it still contains a hamiltonian-chain after deleting any k edges of the hypergraph. What is the minimum number of edges in such a hypergraph? We give lower and upper bounds for this question for several values of r and k. © 2007 Elsevier B.V. All rights reserved.

For a fixed 3-uniform hypergraph F , call a hypergraph F-free if it contains no subhypergraph isomorphic to F . Let ex(n,F) denote the size of a largest F-free hypergraph G ⊆ [n]. Let Fn(F) denote the number of distinct labelled F-free G ⊆ [n]. We show that Fn(F) = 2ex(n,F)+o(n 3), and discuss related problems.

A hypergraph H is called universal for a family F of hypergraphs, if it contains every hypergraph F ∈ F as a copy. For the family of r-uniform hypergraphs with maximum vertex degree bounded by ∆ and at most n vertices any universal hypergraph has to contain Ω(nr−r/∆) many edges. We exploit constructions of Alon and Capalbo to obtain universal r-uniform hypergraphs with the optimal number of edg...

We consider the following definition of connectivity in k-uniform hypergraphs: Two j-sets are j-connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. We determine the threshold at which the random k-uniform hypergraph with edge probability p becomes j-connected with high probability. We also deduce a hitting time result for the rand...

The arboricity of a hypergraph H is the minimum number of acyclic hypergraphs that partition H . The determination of the arboricity of hypergraphs is a problem motivated by database theory. The exact arboricity of the complete k-uniform hypergraph of order n is previously known only for k ∈ {1, 2, n − 2, n − 1, n}. The arboricity of the complete k-uniform hypergraph of order n is determined as...

Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to coregular k-uniform hypergraphs with loops. However, for k ≥ 3 no k-uniform hypergraph is coregular. In this paper we remove the coregular requirement. Consequently, the characterization can be a...

A hypergraph is k-irregular if there is no set of k vertices all of which have the same degree. We asymptotically determine the probability that a random uniform hypergraph is k-irregular.

A tight k-uniform `-cycle, denoted by TC ` , is a k-uniform hypergraph whose vertex set is v0, · · · , v`−1, and the edges are all the k-tuples {vi, vi+1, · · · , vi+k−1}, with subscripts modulo k. Motivated by a classic result in graph theory that every n-vertex cycle-free graph has at most n− 1 edges, Sós and, independently, Verstraëte asked whether for every integer k, a k-uniform n-vertex h...