H-Magic and H-Supermagic of Graphs

An H-covering（resp.decomposition） of a graph G is a set of subgraphs of G,{H;,H;,…,H;} such that H;is isomorphic to H for eachi,and each edge of G belongs to at least（resp.exactly） one of the subgraphs H;,fox 1≤i≤k.An H-covering（resp.decomposition） of a graph G is a magic H-covering（resp.decomposition） if there is a bijection f:V（G）∪E（G）→{1,...,|V（G）|+|E（G）|} such that the sum of labels of edges and the vertices of each copy of H in the decomposition is a constant.If G admits a unique H-covering H and H is a magic H-covering of G,then G is H-magic.In this paper,we show that if G admits a magic H-covering（resp.decomposition）,and satisfies some other conditions,then a union of k vertex joint graph G,kG,and a graph obtained from kG,Gk admit a magic H-covering or decomposition.