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On total matching numbers and total covering numbers for k-uniform hypergraphs. (English) Zbl 0608.05062
Let \(G=(X,E)\) be a graph. The total covering number of G, denoted by \(\alpha_ 2(G)\), is the minimum cardinality of \(| X'| +| E'|\) such that X’\(\subset X\), E’\(\subset E\) and for every \(x\in X\) we have either \(x\in X'\), or \(x\in e\in E'\), or \(x\in jf\in E\) such that \(f\cap X'\neq \emptyset\) and for every \(e\in E\) we have either \(e\in E'\), or \(y\in X'\), \(y\in e\), or \(e\cap f\neq \emptyset\), \(f\in E'\). P. Erdős and A. Meir [Discrete Math. 19, 229-233 (1977; Zbl 0374.05047)] proved inequalities on the total covering (and matching) numbers of complementary graphs. In this paper the author generalizes their results for k-uniform hypergraphs, i.e., for families of k-elements sets. One of his results is: Let \(H=(X,E)\) be a k-uniform hypergraph, \(\bar H\) its complement, \(| X| =n\). Then \[ \lceil (n-1)/k\rceil +1\leq \alpha_ 2(H)+\alpha_ 2(\bar H)\leq \lceil (k+1)n/k\rceil. \]

MSC:
05C65 Hypergraphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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References:
[1] BERGE C.: Graphes et hypergraphes. Dunod, Paris 1970. · Zbl 0213.25702
[2] CHARTRAND G., SCHUSTER S.: On the independence number of complementary graphs. Trans. New York Acad. Sci., 11 36, 1974, 247-251. · Zbl 0275.05110
[3] ERDÖS P., MEIR A.: On total matching numbers and total covering numbers of complementary graphs. Discrete Mathematics 19, 1977, 229-233. · Zbl 0374.05047
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