Hypergraph families with bounded edge cover or transversal number. (English) Zbl 0534.05052

The transversal number \(\tau\), packing number \(\nu\), covering number \(\rho\), and strong stability number \(\alpha\) of hypergraphs are studied. A family F of finite hypergraphs is \(\tau\)-bound (\(\rho\)-bound) if there exists a binding function f such that \(\tau\) (H)\(\leq f(\nu(H))\) (\(\rho\) (H)\(\leq f(\alpha(H)))\) for all \(H\in F\). The existence of a binding function in case of various \(\tau\)-bound (\(\rho\)-bound) is given. It is essentially based on the exclusion of octahedron graphs. Hypergraph versions of the clique color theorem of A. Gyárfás [Infinite finite sets, Colloq. Honour Paul Erdős, Keszthely 1973, Colloq. Math. Soc. János Bolyai 10, 801-816 (1975; Zbl 0307.05111)] are presented. Further results concern Helly hypergraphs with \(C_ 4\)-free line-graph, c-forest hypergraphs, strong Helly hypergraphs, boxes, and polyominoes (cf. the paper of C. Berge, C. C. Chen, V. Chvátal and C. S. Seow [Combinatorica 1, 217-224 (1981; Zbl 0491.05048)]).
Reviewer: J.Plesník


05C65 Hypergraphs
05B40 Combinatorial aspects of packing and covering
Full Text: DOI


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