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Sum labellings of cycle hypergraphs. (English) Zbl 0982.05070

Summary: A hypergraph \({\mathcal H}\) is a sum hypergraph iff there are a finite \(S\subseteq \mathbb{N}^+\) and \(\underline d,\overline d\in \mathbb{N}^+\) with \(1<\underline d\leq\overline d\) such that \({\mathcal H}\) is isomorphic to the hypergraph \({\mathcal H}_{\underline d,\overline d}(S)= (V,{\mathcal E})\) where \(V= S\) and \({\mathcal E}= \{e\subseteq S:\underline d\leq|e|\leq \overline d\wedge \sum_{v\in e}v\in S\}\). For an arbitrary hypergraph \({\mathcal H}\) the sum number \(\sigma= \sigma({\mathcal H})\) is defined to be the minimum number of isolated vertices \(y_1,\dots, y_\sigma\not\in V\) such that \({\mathcal H}\cup \{y_1,\dots, y_\sigma\}\) is a sum hypergraph. Generalizing the graph \(C_n\) we obtain \(d\)-uniform hypergraphs where any \(d\) consecutive vertices of \(C_n\) form an edge. We determine sum numbers and investigate properties of sum labellings for this class of cycle hypergraphs.

MSC:

05C65 Hypergraphs
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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