Expanding Super Edge-Magic Graphs

For a graph G, with the vertex set V(G) and the edge set E(G) an edge- magic total labeling is a bijection f from V to the set of integers with the property that ) ( ) ( G E G ∪ f , , 2 , 1 { " |} ) ( | | ) ( | G E G V + k uv f v f u = + + ) ( ) ( ) ( for each uv ) (G E ∈ and for a fixed integer k. An edge-magic total labeling f is called super edge-magic total labeling if |}