Sparse hypergraphs and pebble game algorithms

Authors: Ileana Streinu and Louis Theran
Journal: European Journal of Combinatorics, 3081944–1964, 2009.
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A hypergraph is -sparse if no subset spans more than hyperedges. We characterize -sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behaviour in terms of the sparsity parameters and . Our constructions extend the pebble games of Lee and Streinu from graphs to hypergraphs.