Magic Labeling of Disjoint Union Graphs

Let G be a graph with vertex set V(G), edge set E(G) and maximum degree Δ 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 $\frac{1+\left|E\left(G\right)\right|}{2}\cdot d\left(v\right)$, 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 ∈ E_v, f(e) ≤ Δ/2}| = |{e : e ∈ E_v, f(e) > Δ/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.