Movatterモバイル変換

[0]ホーム
URL:

Skip to main content

Advertisement

Springer Nature Link
Log in

Distance Magic Labeling of the Halved Foldedn-Cube

  • Conference paper
  • First Online:

Part of the book series:Lecture Notes in Computer Science ((LNCS,volume 13513))

Included in the following conference series:

  • 541Accesses

Abstract

Hypercube is an important structure for computer networks. The distance plays an important role in its applications. In this paper, we study a magic labeling of the halved foldedn-cube which is a variation of then-cube. This labeling is determined by the distance. LetG be a finite undirected simple connected graph with vertex setV(G), distance function\(\partial \) and diameterd. Let\(D\subseteq \{0,1,\dots ,d\}\) be a set of distances. A bijection\(l:V(G)\rightarrow \{1,2,\dots ,|V(G)|\}\) is called aD-magic labeling ofG whenever\(\sum \limits _{x\in G_D(v)}l(x)\) is a constant for any vertex\(v\in V(G)\), where\(G_D(v)=\{x\in V(G): \partial (x,v)\in D\}\). A\(\{1\}\)-magic labeling is also called a distance magic labeling. We show that the halved foldedn-cube has a distance magic labeling (resp. a\(\{0,1\}\)-magic labeling) if and only if\(n=16q^2\)(resp.\(n=16q^2+16q+6\)), whereq is a positive integer.

This is a preview of subscription content,log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 5719
Price includes VAT (Japan)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 7149
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide -see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Similar content being viewed by others

References

  1. Anholcer, M., Cichacz, S., Peterin, I.: Spectra of graphs and closed distance magic labelings. Discrete Math.339(7), 1915–1923 (2016).https://doi.org/10.1016/j.disc.2015.12.025

    Article MathSciNet MATH  Google Scholar 

  2. Anholcer, M., Cichacz, S., Peterin, I., Tepeh, A.: Distance magic labeling and two products of graphs. Graphs Comb.31(5), 1125–1136 (2014).https://doi.org/10.1007/s00373-014-1455-8

    Article MathSciNet MATH  Google Scholar 

  3. Anuwiksa, P., Munemasa, A., Simanjuntak, R.:\(D\)-magic and antimagic labelings of hypercubes.arXiv:1903.05005v2 [math.CO]

  4. Bettayeb, S.: On the\(k\)-ary hypercube. Theor. Comput. Sci.140(2), 333–339 (1995).https://doi.org/10.1016/0304-3975(94)00197-Q

    Article MathSciNet MATH  Google Scholar 

  5. Bloom, G.S., Golomb, S.W.: Applications of numbered undirected graphs. Proc. IEEE65, 562–570 (1977)

    Article  Google Scholar 

  6. Bloom, G.S., Golomb, S.W.: Numbered complete graphs, unusual rulers, and assorted applications. In: Alavi, Y., Lick, D.R. (eds.) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol. 642, pp. 53–65. Springer, Berlin (1978).https://doi.org/10.1007/BFb0070364

    Chapter  Google Scholar 

  7. Bose, B., Broeg, B., Kwon, Y., Ashir, Y.: Lee distance and topological properties of\(k\)-ary\(n\)-cubes. IEEE Trans. Comput.44(8), 1021–1030 (1995)

    Article MathSciNet  Google Scholar 

  8. Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer-Verlag, Berlin (1989).https://doi.org/10.1007/978-3-642-74341-2

    Book MATH  Google Scholar 

  9. Chebotarev, P., Agaev, R.: Matrices of forests, analysis of networks, and ranking problems. Procedia Comput. Sci.17, 1134–1141 (2013)

    Article  Google Scholar 

  10. Cichacz, S., Fronček, D., Krop, E., Raridan, C.: Distance magic cartesian products of graphs. Discuss. Math. Graph Theor.36(2), 299–308 (2016)

    Article MathSciNet  Google Scholar 

  11. Day, K., Al-Ayyoub, A.E.: Fault diameter of\(k\)-ary\(n\)-cube networks. IEEE Trans. Parallel Distrib. Syst.8(9), 903–907 (1997)

    Article  Google Scholar 

  12. Gregor, P., Kovář, P.: Distance magic labelings of hypercubes. Electron. Notes Discrete Math.40, 145–149 (2013).https://doi.org/10.1016/j.endm.2013.05.027

    Article  Google Scholar 

  13. Hsieh, S.Y., Chang, Y.H.: Extraconnectivity of\(k\)-ary\(n\)-cube networks. Theor. Comput. Sci.443(20), 63–69 (2012).https://doi.org/10.1016/j.tcs.2012.03.030

    Article MathSciNet MATH  Google Scholar 

  14. Lin, C.K., Zhang, L.L., Fan, J.X., Wang, D.J.: Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci.634, 97–107 (2016)

    Article MathSciNet  Google Scholar 

  15. Miklavič, Š., Šparl, P.: On distance magic labelings of Hamming graphs and folded hypercubes. Discuss. Math. Graph Theor.0 (2021).https://doi.org/10.7151/dmgt.2430

  16. Miller, M., Rodger, C., Simanjuntak, R.: Distance magic labelings of graphs. Australas. J. Combin.28, 305–315 (2003)

    MathSciNet MATH  Google Scholar 

  17. O’Neal, A., Slater, P.J.: Uniqueness of vertex magic constants. SIAM J. Discrete Math.27(2), 708–716 (2013).https://doi.org/10.1137/110834421

    Article MathSciNet MATH  Google Scholar 

  18. Prajeesh, A.V., Paramasivam, K., Kamatchi, N.: A note on handicap incomplete tournaments. Lect. Notes Comput. Sci. (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)11638, 1–9 (2019).https://doi.org/10.1007/978-3-030-25005-8_1

    Article MathSciNet MATH  Google Scholar 

  19. Sedláček, J.: Some properties of interchange graphs. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), pp. 145–150. House Czech. Acad. Sci., Prague (1964)

    Google Scholar 

  20. Simanjuntak, R., Anuwiksa, P.:\(D\)-magic strongly regular graphs. AKCE Int. J. Graphs Comb.17(3), 995–999 (2020)

    Article MathSciNet  Google Scholar 

  21. Simó, E., Yebra, J.L.A.: The vulnerability of the diameter of folded\(n\)-cubes. Discrete Math.174(1–3), 317–322 (1997)

    Article MathSciNet  Google Scholar 

  22. Tian, Y., Hou, L.H., Hou, B., Gao, S.G.:\(D\)-magic labelings of the folded\(n\)-cube. Discrete Math.344(9), 112520 (2021)

    Article MathSciNet  Google Scholar 

  23. van Dam, E.R., Koolen, J.H., Tanaka, H.: Distance-regular graphs. Electron. J. Combin. DS22 (2016)

    Google Scholar 

  24. Vilfred, V.: Sigma labelled graphs and circulant graphs. Ph.D. Thesis, University of Kerala, (1994)

    Google Scholar 

  25. Zhao, S.L., Yang, W.H., Zhang, S.R.: Component connectivity of hypercubes. Theor. Comput. Sci.640(C), 115–118 (2016).https://doi.org/10.1016/j.tcs.2016.05.035

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant 11971146) and the National Natural Science Foundation of Hebei Province (Grant A2017403010).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Yi Tian & Suogang Gao

  2. School of Big Data Science, Hebei Finance University, Baoding, 071051, People’s Republic of China

    Yi Tian

  3. School of Mathematics and Science, Hebei GEO University, Shijiazhuang, 050024, People’s Republic of China

    Na Kang

  4. Department of Computer Science, University of Texas at Dallas, Richardson, TX, 75080, USA

    Weili Wu & Ding-Zhu Du

  5. Hebei International Joint Research Center for Mathematics and Interdisciplinary Science, Shijiazhuang, 050024, People’s Republic of China

    Suogang Gao

Authors
  1. Yi Tian

    You can also search for this author inPubMed Google Scholar

  2. Na Kang

    You can also search for this author inPubMed Google Scholar

  3. Weili Wu

    You can also search for this author inPubMed Google Scholar

  4. Ding-Zhu Du

    You can also search for this author inPubMed Google Scholar

  5. Suogang Gao

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence toSuogang Gao.

Editor information

Editors and Affiliations

  1. Guangdong University of Technology, Guangzhou, China

    Qiufen Ni

  2. University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

Rights and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Tian, Y., Kang, N., Wu, W., Du, DZ., Gao, S. (2022). Distance Magic Labeling of the Halved Foldedn-Cube. In: Ni, Q., Wu, W. (eds) Algorithmic Aspects in Information and Management. AAIM 2022. Lecture Notes in Computer Science, vol 13513. Springer, Cham. https://doi.org/10.1007/978-3-031-16081-3_28

Download citation

Publish with us

Access this chapter

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 5719
Price includes VAT (Japan)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 7149
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide -see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

[8]ページ先頭
©2009-2025 Movatter.jp