Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees

  • Vladimir Blinovsky
  • Catherine Greenhill
Keywords: Hypergraph, Asymptotic enumeration, Switching

Abstract

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For n3, let r=r(n)3 be an integer and let \boldsymbol{k} = (k_1,\ldots, k_n) be a vector of nonnegative integers, where each k_j = k_j(n) may depend on n. Let M = M(n) = \sum_{j=1}^n k_j for all n\geq 3, and define the set \mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}. We assume that \mathcal{I} is infinite, and perform asymptotics as n tends to infinity along \mathcal{I}. Our main result is an asymptotic enumeration formula for linear r-uniform hypergraphs with degree sequence \boldsymbol{k}. This formula holds whenever the maximum degree k_{\max} satisfies r^4 k_{\max}^4(k_{\max} + r) = o(M). Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.
Published
2016-08-05
Article Number
P3.17