On H-supermagic labelings for certain shackles and amalgamations of a connected graph

Let H be a graph. A graph G = (V,E) is said to be H-magic if every edge of G belongs to at least one subgraph isomorphic to H and there is a total labeling f:V ⋃ E → {1,2, . . . ,|V|+ |E|} such that for each subgraph H' = (V',E') of G isomorphic to H, the sum of all vertex labels in V' plus the sum of all edge labels in E' is a fixed constant. Additionally, G is said to be H-supermagic if f(V) = {1,2, .. . , |V|}. We study H-supermagic labelings of some graphs obtained from k isomorphic copies of a connected graph H. By using a k-balanced partition of multisets, we prove that certain shackles and amalgamations of a connected graph H are H-supermagic