Acta Mathematica Sinica, English Series

, Volume 35, Issue 11, pp 1817–1826

# Magic Labeling of Disjoint Union Graphs

• Tao Wang
• Ming Ju Liu
• De Ming Li
Article

## Abstract

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 $${{1 + \left| {E(G)} \right|} \over 2} \cdot d(v)$$, where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex vV(G), |{e : eEv, f(e) ≤ Δ/2}| = |{e : eEv, 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.

## Keywords

Supermagic graphs degree-magic graphs strong vertex-magic edge coloring

05C78

## Notes

### Acknowledgements

We would like to thank the anonymous referee for reading of our manuscript and for invaluable comments.

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