For r ≥ 2, an r-uniform hypergraph is called a friendship r-hypergraph if every set R of r vertices has a unique ‘friend’ – that is, there exists a unique vertex x / ∈ R with the property that for each subset A ⊆ R of size r − 1, the set A ∪ {x} is a hyperedge. We show that for r ≥ 3, the number of hyperedges in a friendship r-hypergraph is at least r+1 r ( n−1 r−1 ) , and we characterise those...

Erdős, Rubin, and Taylor found a nice correspondence between the minimum order of a complete bipartite graph that is not r-choosable and the minimum number of edges in an r-uniform hypergraph that is not 2-colorable (in the ordinary sense). In this note we use their ideas to derive similar correspondences for complete kpartite graphs and complete k-uniform k-partite hypergraphs.

Covering arrays are combinatorial objects that have been successfully applied in the design of test suites for testing systems such as software, circuits and networks, where failures can be caused by the interaction between their parameters. In this paper, we perform a new generalization of covering arrays called covering arrays on 3-uniform hypergraphs. Let n, k be positive integers with k ≥ 3...

In this article, we investigate the sensitivity complexity of hypergraph properties. We present a k -uniform property with O ( n (⌈ k/3 ⌉) for any ≥ 3 , where is number vertices. Moreover, can do better when ≡ 1 (mod 3) by presenting (n⌈ ⌉-1/2). This result disproves conjecture Babai, which conjectures that properties at least Ω k/2 ). also other symmetric functions and show many classes transi...

One of the most important recent developments in the complexity of approximate counting is the classification of the complexity of approximating the partition functions of antiferromagnetic 2-spin systems on bounded-degree graphs. This classification is based on a beautiful connection to the so-called uniqueness phase transition from statistical physics on the infinite ∆-regular tree. Our objec...

For a hypergraph G with v vertices and e~ edges of size i, the average vertex degree is d(G) = =Eiedv. Call balanced if d(H)~-d(G) for all subhyoergraphs H of G. Let re(G0= max d(H). Hc=G A hypergraph F is said to be a balanced extension of G if GC F, F is balanced and tifF)-re(G), i.e. F is balanced and does not increase the maximum average degree. It is shown that for every hypergraph G there...

Using the symmetric form of the Lovász Local Lemma, one can conclude that a k-uniform hypergraph H admits a proper 2-colouring if the maximum degree (denoted by ∆) of H is at most 2 8k independently of the size of the hypergraph. However, this argument does not give us an algorithm to find a proper 2-colouring of such hypergraphs. We call a hypergraph linear if no two hyperedges have more than ...

An h-uniform hypergraph (h ≥ 2) H = (V, E) of order n = |V| and size m = |E|, consists of a vertex set V(H) = V and edge set E(H) = E , where E ⊂ V and |E| = h for each edge E in E . H is said to be linear if 0 ≤ |E ∩ F | ≤ 1 for any two distinct edges E,F ∈ E(H) [1]. Let P h,1 p denote the linear path consisting of p ≥ 1 edges E1, . . . , Ep such that |E1| = . . . = |Ep| = h, |Ek ∩ El| = 1 if ...

Let H=(V,E) be a hypergraph. A panchromatic t-colouring of H is a t-colouring of its vertices such that each edge has at least one vertex of each colour; and H is panchromatically t-choosable if, whenever each vertex is given a list of t colours, the vertices can be coloured from their lists in such a way that each edge receives at least t different colours. The Hall ratio of H is h(H) =min {∣∣...

Let H be a 4-uniform hypergraph on n vertices. The transversal number τ(H) of H is the minimum number of vertices that intersect every edge. The result in [J. Combin. Theory Ser. B 50 (1990), 129–133] by Lai and Chang implies that τ(H) 6 7n/18 when H is 3-regular. The main result in [Combinatorica 27 (2007), 473–487] by Thomassé and Yeo implies an improved bound of τ(H) 6 8n/21. We provide a fu...