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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Berge,Graphs and hypergraphs, North-Holland (1973).
B. Bollobás, On complete subgraphs of different orders,Math. Proc. Cambr. Philos. Soc. 79 (1976) 19–24.
B. Bollobás, Extremal problems in graph theory,J. Graph Theory 1 (1977) 117–123.
P. Erdős, A. W. Goodman andL. Pósa, The representation of a graph by set intersections,Canad. J. Math. 18 (1966) 106–112.
P. Erdős, On some extremal problems in graph theory,Israel Journ. of Math. 3 (1965) 113–116.
P. Erdős, On the combinatorial problems which I would most like to see solved,Combinatorica 1 (1981) 25–42.
T. Gallai, Über extreme Punkt und Kantenmengen,Ann. Univ. Sci. Budapest Eötvös Sect. Math. 2 (1959) 133–138.
J. Lehel andZs. Tuza, Triangle-free partial graphs and edge convering theorems,Discrete Math. 39 (1982), 59–65.
Author information
Authors and Affiliations
Additional information
Dedicated to Tibor Gallai on his seventieth birthday
Rights and permissions
About this article
Cite this article
Lehel, J. Covers in hypergraphs. Combinatorica 2, 305–309 (1982). https://doi.org/10.1007/BF02579237
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02579237