Cores of random r-partite hypergraphs

Authors:
Fabiano C. Botelho;Nicholas Wormald;Nivio Ziviani
Affiliations:
Data Domain an EMC company, Santa Clara, CA, USA and Department of Computer Science, Federal University of Minas Gerais, Belo Horizonte, Brazil;Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada;Department of Computer Science, Federal University of Minas Gerais, Belo Horizonte, Brazil
Venue:
Information Processing Letters
Year:
2012

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Abstract

We show that the threshold c"r","k (in terms of the average degree of the graph) for appearance of a k-core in a random r-partite r-uniform hypergraph G"r","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. The case r,k=2 was analyzed in Botelho et al. (2007) [4] but the general case r=3, k=2 is still open. Besides the proof for the general case, we have also provided a simpler proof for the case r,k=2. This problem was provided without a proof (but with strong experimental evidence) in the analysis of the algorithm presented in Botelho et al. (2007) [2]. This 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. For an input key set S with m keys, the family of minimal perfect hash functions generated by the algorithm can be stored in O(m) bits, where the hidden constant is within a factor of two from the information theoretical lower bound.