Magic Labeling of Disjoint Union Graphs

摘要：Let G be a graph with vertex set V(G),edge set E(G)and maximum degreeΔrespectively.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 to1+|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)≤Δ/2}|=|{e:e∈Ev,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 n G is a d-magic graph.Results in this paper generalise some known results.