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 is an element of E to one of its vertices v is an element of e such that no vertex is assigned more than l edges. Let H-n,H-m,H-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 H-n,H-m,H-k for all k >= 3 and l >= 2, that is, we determine a critical quantity c(*)k,l such that with probability 1-o(1) the graph H-n,H-cn,(k) has an l-orientation if c c(k,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