## On the minimum size of tight hypergraphs.(English)Zbl 0776.05079

Summary: A $$k$$-graph, $$H=(V,E)$$, is tight if for every surjective mapping $$f:V\to\{1,\dots,k\}$$ there exists an edge $$\alpha\in E$$ such that $$f|_ \alpha$$ is injective. Clearly, 2-graphs are tight if and only if they are connected. Bounds for the minimum number $$\varphi^ k_ n$$ of edges in a tight $$k$$-graph with $$n$$ vertices are given. We conjecture that $$\varphi^ 3_ n=\lceil n(n-2)/3\rceil$$ for every $$n$$ and prove the equality when $$2n+1$$ is prime. From the examples, minimal embeddings of complete graphs into surfaces follow.

### MSC:

 05C65 Hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C35 Extremal problems in graph theory 05C15 Coloring of graphs and hypergraphs 05C05 Trees

### Keywords:

tight hypergraphs; minimal embeddings
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### References:

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