Covers in hypergraphs

Abstract

Let α(H) denote the stability number of a hypergraph H. The covering number ρ{variant}(H) is defined as the minimal number of edges from H to cover its vertex set V(H). The main result is the following extension of König's wellknown theorem: If α(H′)≧|V(H′)|/2 holds for every section hypergraph H′ of H then ρ{variant}(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. © 1982 Akadémiai Kiadó.