Antimagic orientations for the complete k-ary trees

  • Chen Song
  • Rong-Xia HaoEmail author


A labeling of a digraph D with m arcs is a bijection from the set of arcs of D to \(\{1,2,\ldots ,m\}\). A labeling of D is antimagic if all vertex-sums of vertices in D are pairwise distinct, where the vertex-sum of a vertex \(u \in V(D)\) for a labeling is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. Hefetz et al. (J Graph Theory 64:219–232, 2010) conjectured that every connected graph admits an antimagic orientation. We support this conjecture for the complete k-ary trees and show that all the complete k-ary trees \(T_k^r\) with height r have antimagic orientations for any k and r.


Complete k-ary tree Antimagic labeling Antimagic orientation 



This work was supported by the National Natural Science Foundation of China (No. 11731002), the Fundamental Research Funds for the Central Universities (Nos. 2016JBM071, 2016JBZ012).


  1. Alon N, Kaplan G, Lev A, Roditty T, Yuster R (2004) Dense graphs are antimagic. J Graph Theory 47:297–309MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chang F, Liang Y-C, Pan Z, Zhu X (2016) Antimagic labeling of regular graphs. J Graph Theory 82:339–349MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cranston DW (2009) Regular bipartite graphs are antimagic. J Graph Theory 60:173–182MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cranston DW, Liang Y-C, Zhu X (2015) Regular graphs of odd degree are antimagic. J Graph Theory 80:28–33MathSciNetCrossRefzbMATHGoogle Scholar
  5. Hefetz D, Mütze T, Schwartz J (2010) On antimagic directed graphs. J Graph Theory 64:219–232MathSciNetzbMATHGoogle Scholar
  6. Li T, Song Z-X, Wang GH, Yang DL, Zhang C-Q (2019) Antimagic orientations of even regular graphs. J Graph Theory 90:46–53MathSciNetCrossRefzbMATHGoogle Scholar
  7. Lozano A, Mora M, Seara C (2019) Antimagic labelings of caterpillars. Appl Math Comput 347:734–740MathSciNetGoogle Scholar
  8. Shan S, Yu X (2017) Antimagic orientation of biregular bipartite graphs. Electron J Comb 24(4). #P4.31Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Jiaotong UniversityBeijingPeople’s Republic of China

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