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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
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