Super Edge-Magic Labeling of 5-Uniform and 6-Uniform Hypergraphs Generated by Arbitrary Simple Graphs: Authawich Narissayaporn, Ratinan Boonklurb, Sirirat Singhun - Team, Support (Thai Journal of Mathematics, 2020)
The super edge-magic (SEM) labeling on hypergraphs is the extension of the SEM labeling on graphs. For a hypergraph H with vertex set V_H and hyperedge set E_H, we call a bijective mapping f:V_H cup E_H to 1,2,3,…,|V_H|+|E_H|} as an SEM labeling of H if and only if (i) there is an integer Λ such that for every eE_H, f(e)+_ve f(v)=Λ and (ii) f(V_H)={1,2,3,…,|V_H|}. In this article, we define 5-uniform H^(5)(G) and 6-uniform H^(6)(G) hypergraphs from an arbitrary simple graph G and show that H^(5)(G) is always an SEM hypergraph. However, if G has odd number of edges, then H^(6)(G) is an SEM hypergraph. Unfortunately, if G has even number of edges, the SEM labeling for H^(6)(G) depends on the structure of the hypergraph. Thus, an example of SEM labeling of H^(6)(nC_4), which has even number of edges, is given. Finally, if H is a k-uniform SEM hypergraph, then we can show that H', obtained from H by adding more vertices, is (k+2m)-uniform SEM hypergraph.