Sum labellings of cycle hypergraphs

A hypergraph is a sum hypergraph iff there are a finite S ⊆ IN⁺ and d̲, [d̅] ∈ IN⁺ with 1 < d̲ ≤ [d̅] such that is isomorphic to the hypergraph ${\displaystyle \_\{d̲,[d̅]\}\left(S\right)=(V,)}$ where V = S and ${\displaystyle =\{e\subseteq S:d̲\le \left|e\right|\le [d̅]\wedge {\sum }_{v\in e}v\in S\}}$. For an arbitrary hypergraph the sum number σ = σ() is defined to be the minimum number of isolated vertices ${\displaystyle y₁,...,y_{\sigma }\notin V}$ such that ${\displaystyle \cup \{y₁,...,y_{\sigma }\}}$ 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.