Super edge-magic labeling for 𝒌-uniform, complete 𝒌-uniform and complete 𝒌-uniform 𝒌-partite hypergraphs

Abstract

Let 𝐻 be a hypergraph with a vertex set 𝑉 and a hyperedge set 𝐸. Generalized from the super edge-magic in a graph, we say that a hypergraph 𝐻 is super edge-magic if there is a bijection 𝑓: 𝑉 ∪ 𝐸 → {1,2,3, … , |𝑉| + |𝐸|} which satisfies: (i) there exists a constant Λ such that for all 𝑒 ∈ 𝐸, 𝑓(𝑒) + ∑𝑣∈𝑒 𝑓(𝑣) = Λ and (ii) 𝑓(𝑉) = {1,2,3, … , |𝑉|}. In this paper, we give a necessary condition for a 𝑘-uniform hypergraph to be super edge-magic. We show that the complete 𝑘-uniform hypergraph of 𝑛 vertices is super edge-magic if and only if 𝑘 ∈ {0,1, 𝑛 − 1, 𝑛}. Finally, we also prove that the complete 𝑘-uniform 𝑘-partite hypergraph with the same number of vertices in each partite, namely 𝑛, is super edge-magic if and only if (𝑛, 𝑘) = (1, 𝑘) for all 𝑘 ≥ 2 and (𝑛, 𝑘) = (2, 3).

Keywords

• super edge-magic
• complete 𝑘-uniform hypergraph
• complete 𝑘-uniform 𝑘-partite hypergraph
• hypergraph
• labeling