Nonconvexity of the set of hypergraph degree sequences

It is well known that the set of possible degree sequences for a graph on $n$ vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a $k$-uniform hypergraph on $n$ vertices is not the intersection of a lattice and a convex polytope for $k \geq 3$ and $n \geq k+13$. We also show an analogous nonconvexity result for the set of degree sequences of $k$-partite $k$-uniform hypergraphs and the generalized notion of $\lambda$-balanced $k$-uniform hypergraphs.Comment: 5 page

Sampling uniform hypergraphs with given degrees

Graphs are combinatorial objects commonly used to model relationships between pairs of entities. Hypergraphs are a generalization of graphs in which edges connect an arbitrary number of vertices. We consider hypergraphs in which each edge has size k, each vertex has a degree specified by a degree sequence d, and all edges are unique. These are known as simple k-uniform hypergraphs with degree sequence d. We focus on algorithms for sampling these hypergraphs, particularly when the degree sequence is approximately regular and sufficiently sparse. The goal is an algorithm which produces approximately uniform output with expected running time that is polynomial in the number of vertices. We first discuss an algorithm for this problem which used a rejection sampling approach and a black-box bipartite graph sampler. This algorithm was presented in a paper by myself and co-authors: my specific contributions to the publication are described. As a new contribution (not contained in the paper), the rejection sampling approach is extended to give an algorithm for sampling linear hypergraphs, which are hypergraphs in which no two distinct edges share more than one common vertex. We also define and analyse an algorithm for sampling simple k-uniform hypergraphs with degree sequence d. Our algorithm uses a black-box sampler A for producing (possibly non-simple) hypergraphs and a ‘switchings’ process to remove any repeated edges from the hypergraph. This analysis additionally produces explicit tail bounds for the number and multiplicity of repeated edges in uniformly distributed random hypergraphs, under certain conditions for d and k. We show that our algorithm is asymptotically approximately uniform and has an expected running time that is polynomial in the number of vertices for a large range of degree sequences d, provided d is near-regular. This extends the range of degree sequences for which efficient sampling schemes are known

On the Reconstruction of 3-Uniform Hypergraphs from Degree Sequences of Span-Two

A nonnegative integer sequence is k-graphic if it is the degree sequence of a k-uniform simple hypergraph. The problem of deciding whether a given sequence π is 3-graphic has recently been proved to be NP-complete, after years of studies. Thus, it acquires primary relevance to detect classes of degree sequences whose graphicality can be tested in polynomial time in order to restrict the NP-hard core of the problem and design algorithms that can also be useful in different research areas. Several necessary and few sufficient conditions for π to be k-graphic, with k≥ 3 , appear in the literature. Frosini et al. defined a polynomial time algorithm to reconstruct k-uniform hypergraphs having regular or almost regular degree sequences. Our study fits in this research line providing a combinatorial characterization of span-two sequences, i.e., sequences of the form π= (d, … , d, d- 1 , … , d- 1 , d- 2 , … , d- 2 ) , d≥ 2 , which are degree sequences of some 3-uniform hypergraphs. Then, we define a polynomial time algorithm to reconstruct one of the related 3-uniform hypergraphs. Our results are likely to be easily generalized to k≥ 4 and to other families of degree sequences having simple characterization, such as gap-free sequences

Degree sequences of sufficiently dense random uniform hypergraphs

We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edges. More specifically, we compare the degree sequence of a random $r$-uniform hypergraph with a given number edges to certain models involving sequences of binomial or hypergeometric random variables conditioned on their sum

Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs

The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that, in fact, every positive independence density of a countably infinite hypergraph with hyperedges of bounded size is equal to the independence density of some finite hypergraph whose hyperedges are no larger than those in the infinite hypergraph. This answers a question of Bonato, Brown, Kemkes, and Pra{\l}at about independence densities of graphs. Furthermore, we show that for any $k$, the set of independence densities of hypergraphs with hyperedges of size at most $k$ is closed and contains no infinite increasing sequences.Comment: To appear in the European Journal of Combinatorics, 12 page