Cores of random r-partite hypergraphs

We show that the threshold cr,k for appearance of a k-core in a random r-partite r-uniform hypergraph Gr,n,m is the same as for a random r-uniform hypergraph with cn/r edges without the r-partite restriction, where r, k ≥ 2. In both cases, the average degree is c. This is an important problem in the analysis of the algorithm presented in [2]. The algorithm constructs a family of minimal perfect hash functions based on random r-partite r-uniform hypergraphs with an empty k-core subgraph, for k ≥ 2. The above claim was not proved but was provided with strong experimental evidence. For an input key set S with m keys, the algorithm was the first one capable of constructing a simple and efficient family of minimal perfect hash functions that can be stored in O(m) bits, where the hidden constant is within a factor of two from the information theoretical lower bound. The case r, k = 2 was analyzed in [3] but the general case r ≥ 3, k ≥ 2 was still open.