A new approach to the orientation of random hypergraphs

A h-uniform hypergraph H = (V,E) is called (l, k)-orientable if there exists an assignment of each hyperedge e ∈ E to exactly l of its vertices v ∈ e such that no vertex is assigned more than k hyperedges. Let Hn,m,h be a hypergraph, drawn uniformly at random from the set of all h-uniform hypergraphs with n vertices and m edges. In this paper, we determine the threshold of the existence of a (l, k)-orientation of Hn,m,h for k ≥ 1 and h > l ≥ 1, extending recent results motivated by applications such as cuckoo hashing or load balancing with guaranteed maximum load. Our proof combines the local weak convergence of sparse graphs and a careful analysis of a Gibbs measure on spanning subgraphs with degree constraints. It allows us to deal with a much broader class than the uniform hypergraphs.