# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
  BERGE C.: Graphes et hypergraphes. Dunod, Paris 1970. · Zbl 0213.25702  CHARTRAND G., SCHUSTER S.: On the independence number of complementary graphs. Trans. New York Acad. Sci., 11 36, 1974, 247-251. · Zbl 0275.05110  ERDÖS P., MEIR A.: On total matching numbers and total covering numbers of complementary graphs. Discrete Mathematics 19, 1977, 229-233. · Zbl 0374.05047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.