Supermagic coverings of the disjoint union of graphs and amalgamations

Let HH be a graph. A graph G=(V,E)G=(V,E) admits an HH-covering   if every edge in EE belongs to a subgraph of GG isomorphic to HH. A graph GG is called HH-magic   if there is a fixed integer kk and a total labeling f:V∪E→{1,2,…,|V|+|E|}f:V∪E→{1,2,…,|V|+|E|} such that for each subgraph H′=(V′,E′)H′=(V′,E′) of GG isomorphic to HH, ∑v∈V′f(v)+∑e∈E′f(e)=k∑v∈V′f(v)+∑e∈E′f(e)=k. If f(V)={1,2,…,|V|}f(V)={1,2,…,|V|}, then GG is HH-supermagic  . In this paper, we investigate the GG-supermagicness of a disjoint union of cc copies of a graph GG. We characterize all such graphs of being GG-supermagic. We also show that a disjoint union of any paths is cPhcPh-supermagic for some cc and hh. Besides, we prove that certain subgraph  -amalgamation of graphs GG is GG-supermagic.