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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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