We present results from two papers by the authors on analysis of d-regular k-uniform hypergraphs, when k is fixed and the number n of vertices tends to infinity. The first result is approximate enumeration of such hypergraphs, provided d = d(n) = o(nκ), where κ = κ(k) = 1 for all k ≥ 4, while κ(3) = 1/2. The second result is that a random d-regular hypergraph contains as a dense subgraph the un...

Let 0 be any fixed 3-uniform hypergraph. For a 3-uniform hypergraph we define 0( ) to be the maximum size of a set of pairwise triple-disjoint copies of 0 in . We say a function from the set of copies of 0 in to [0, 1] is a fractional 0-packing of if ¥ e ( ) 1 for every triple e of . Then * 0( ) is defined to be the maximum value of ¥ 0 over all fractional 0-packings of . We show that * 0( ) 0(...

A graph G is a support for a hypergraph H = (V,S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si ∈ S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [11] proved that it is NPcomplete to decide if a given hypergraph has a planar support. In contrast, there are lienar time algorithms to tes...

Let H be a hypergraph and let Lv : v ∈ V (H) be sets; we refer to these sets as lists and their elements as colors. A list coloring of H is an assignment of a color from Lv to each v ∈ V (H) in such a way that every edge of H contains a pair of vertices of different colors. The hypergraph H is k-list-colorable if it has a list coloring from any collection of lists of size k. The list chromatic ...

The simple group F of order 168 which first occurred in the work of Galois around 1830 reappeared in two geometric contexts later in the 19th century. It arose in Felix Klein's investigation in 1877 of a Riemann surface of genus 3 which admits F as its group of conformal homeomorphisms. This is the least genus for which Hurwitz's 84(g—1) bound is attained. (See [6] for an interesting historical...

A mixed hypergraph H is a triple (V, C,D) where V is its vertex set and C and D are families of subsets of V , C–edges and D–edges. A mixed hypergraph is a bihypergraph iff C = D. A hypergraph is planar if its bipartite incidence graph is planar. A vertex coloring of H is proper if each C–edge contains two vertices with the same color and each D–edge contains two vertices with different colors....

Given a regular action of a nite group G on a set V , we ask (and answer) the question of the existence of an incidence structure I = (V; B) on the set V whose full automorphism group Aut(I) is the group G in its regular action. Additional conditions on I also allow us to reene the original problem to the class of hypergraphs. Using results on graphical and digraphical regular representations (...

Abstract Most opinion dynamics models are based on pairwise interactions. However in many real situations, discussions take place within groups of people. Here, we define a higher order Deffuant model by generalizing the original interaction for bounded-confidence opinion-dynamics to interactions involving group agents size k . The generalized is naturally encoded hypergraph. We study this diff...

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph pLaplacian. Unlike the existing two-node graph Laplacians, this generalization makes it possible to ana...

We prove that a hypergraph is a product of a finite number of edges if and only if it is interval-regular, satisfies the gated-edge property and has a vertex of finite degree. As a consequence, we get a characterization of Hamming graphs.