Acta Mathematica Sinica, English Series

, Volume 35, Issue 11, pp 1817–1826 | Cite as

Magic Labeling of Disjoint Union Graphs

  • Tao WangEmail author
  • Ming Ju Liu
  • De Ming Li


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.


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

MR(2010) Subject Classification



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


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Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.College of ScienceNorth China Institute of Science and TechnologyLangfangP. R. China
  2. 2.Department of MathematicsBeihang UniversityBeijingP. R. China
  3. 3.LMIB of the Ministry of EducationBeijingP. R. China
  4. 4.Department of MathematicsCapital Normal UniversityBeijingP. R. China

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