Sum labellings of cycle hypergraphs

A hypergraph H is a sum hypergraph iff there are a finite S ⊆ IN and d, d ∈ IN with 1 < d ≤ d such that H is isomorphic to the hypergraph Hd,d(S) = (V, E) where V = S and E = {e ⊆ S : d ≤ |e| ≤ d ∧ v∈e v ∈ S}. For an arbitrary hypergraph H the sum number σ = σ(H) is defined to be the minimum number of isolated vertices y1, . . . , yσ 6∈ V such that H ∪ {y1, . . . , yσ} is a sum hypergraph. Generalizing the graph Cn we obtain d-uniform hypergraphs where any d consecutive vertices of Cn form an edge. We determine sum numbers and investigate properties of sum labellings for this class of cycle hypergraphs.