Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees

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 n ≥ 3, let r = r(n) ≥ 3 be an integer and let k = (k1, . . . , kn) be a vector of nonnegative integers, where each kj = kj(n) may depend on n. Let M = M(n) = ∑n j=1 kj for all n ≥ 3, and define the set I = {n ≥ 3 | r(n) divides M(n)}. We assume that I is infinite, and perform asymptotics as n tends to infinity along I. Our main result is an asymptotic enumeration formula for linear r-uniform hypergraphs with degree sequence k. This formula holds whenever the maximum degree kmax satisfies r 4k4 max(kmax+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. ∗Research supported by the Australian Research Council Discovery Project DP140101519.