H-magic covering on some classes of graphs

For a graph G(V,E), an edge-covering of G is a family of different subgraphs H₁,…H_k such that any edge of E belongs to at least one of the subgraphs H_i, 1 ≤ i ≤ k. If every H_i is isomorphic to a given graph H, then G admits an H-covering. Graph G is said to be H-magic if G has an H-covering 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,$ΣνϵV′f(ν)+ΣeϵE′f(e)$ is fixed constant. Furthermore, if $f(V) = }1,2,…|V|{$ then G is called H-supermagic. The sum of all vertex and edge labels on H, under a labeling f, is denoted by Σf(H). In this paper we study H-supermagic labeling for some classes of trees such as a double star, a caterpillar, a firecracker and a banana tree.