The multiple-orientability thresholds for random hypergraphs

Abstract

A k-uniform hypergraph H = (V, E) is called l-orientable, if there is an assignment of each edge e ∈ E to one of its vertices ν ∈ e such that no vertex is assigned more than l edges. Let Hn,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the l-orientability of Hn,m,k for all k ≥ 3 and l ≥ 1, i.e., we determine a critical quantity ck, l* such that with probability 1 - o(1) the graph Hn,cn,k has an l- orientation if c < c k, l* but fails doing so if c > ck, l*. Our result has various applications including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.