Covers in hypergraphs

Abstract

Let α(H) denote the stability number of a hypergraphH. The covering number ϱ(H) is defined as the minimal number of edges fromH to cover its vertex setV(H). The main result is the following extension of König’s wellknown theorem:

If α(H′)≧|V(H′)|/2 holds for every section hypergraphH′ ofH then ϱ(H)≦α(H).

This theorem is applied to obtain upper bounds on certain covering numbers of graphs and hypergraphs. In par ticular, we prove a conjecture of B. Bollobás involving the hypergraph Turán numbers.

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Dedicated to Tibor Gallai on his seventieth birthday

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Lehel, J. Covers in hypergraphs. Combinatorica 2, 305–309 (1982). https://doi.org/10.1007/BF02579237

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AMS subject classification (1980)

  • 05 C 65