Polychromatic colorings and cover decompositions of hypergraphs

摘要

A polychromatic coloring of a hypergraph is a coloring of its vertices in such a way that every hyperedge contains at least one vertex of each color. A polychromatic m-coloring of a hypergraph H corresponds to a cover m-decomposition of its dual hypergraph H*. The maximum integer m that a hypergraph H admits a cover m-decomposition is exactly the longest lifetime for a wireless sensor network (WSN) corresponding to the hypergraph H. In this paper, we show that every hypergraph H has a polychromatic m-coloring if m≤⌊Sln(cΔS2)⌋, where 0 < c < 1, and Δ ≥ 1, S ≥ 2 are the maximum degree, the minimum size for all hyperedges in H, respectively. This result improves a result of Henning and Yeo on polychromatic colorings of hypergraphs in 2013, and its dual form improves one of Bollobás, Pritchard, Rothvoß, and Scott on cover decompositions of hypergraphs in 2013. Furthermore, we give a sufficient condition for a hypergraph H to have an “equitable” polychromatic coloring, which extends the result of Henning and Yeo in 2013 and improves in part one of Beck and Fiala in 1981 on 2-colorings (property B) of hypergraphs.