Magic Labeling of Disjoint Union Graphs

Let G be a graph with vertex set V(G), edge set E(G) and maximum degree . respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, ., |E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to 1+|E(G)|2.d(v), where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v . V(G), |{e : e . Ev, f(e) . ./2}| = |{e : e . Ev, f(e) > ./2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m . 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n . 2, then nG is a d-magic graph. Results in this paper generalise some known results.