Strong labelings of linear forests

  • Martin Bača
  • Yu Qing Lin
  • Francesc A. Muntaner-Batle
  • Miquel Rius-Font
Article

Abstract

A (p, q)-graph G is called super edge-magic if there exists a bijective function f: V (G) ∪ E(G) → {1, 2, ..., p+q} such that f(u)+f(υ)+f() is a constant for each ε E(G) and f(V (G)) = {1, 2, ..., p}.

In this paper, we introduce the concept of strong super edge-magic labeling as a particular class of super edge-magic labelings and we use such labelings in order to show that the number of super edge-magic labelings of an odd union of path-like trees (mT), all of them of the same order, grows at least exponentially with m.

Keywords

linear forest path-like tree strong super edge magic labeling 

MR(2000) Subject Classification

05C05 05C78 

References

  1. [1]
    Chartrand, G., Lesniak, L.: Graphs and Digraphs, Monterey: Wadsworth & Brooks/Cole Advanced Books and Software, 1986Google Scholar
  2. [2]
    Gallian, J.: A dynamic survey of graph labeling. Electron. J. Combin., 15(#DS6), (2007)Google Scholar
  3. [3]
    Tan, Y. Y., Fan, Y. Z.: On edge singularity and eigenvectors of mixed graphs. Acta Mathematica Sinica, English Series, 24(1), 139–146 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Shiu, W. C., Liu, G. Z.: k-factors in regular graphs. Acta Mathematica Sinica, English Series, 24(7), 1213–1220 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Shan, E. F., Sohn, M. Y., Yuan, X. D., et al.: Domination number in graphs with minimum degree two. Acta Mathematica Sinica, English Series, 25(8), 1253–1268 (2009)MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Kotzig, A., Rosa, A.: Magic valuations of finite graphs. Canad. Math. Bull., 13, 451–461 (1970)MATHMathSciNetGoogle Scholar
  7. [7]
    Ringel, G., Lladó, A. S.: Another tree conjecture. Bull. Inst. Combin. Appl., 18, 83–85 (1996)MATHMathSciNetGoogle Scholar
  8. [8]
    Wallis, W. D.: Magic Graphs, Birkhäuser, Boston, 2001MATHGoogle Scholar
  9. [9]
    Enomoto, H., Lladó, A.S., Nakamigawa, T., Ringel, G.: Super edge-magic graphs. SUT J. Math., 34, 105–109 (1998)MATHMathSciNetGoogle Scholar
  10. [10]
    Figueroa-Centeno, R. M., Ichishima, R., Muntaner-Batle, F. A.: The place of super edge-magic labelings among other classes of labelings. Discrete Math., 231, 153–168 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Rosa, A.: On certain valuations of the vertices of a graph. In: Gordon and Breach (Ed.), Theory of Graphs, Dunod, Paris, 1967, 349–355Google Scholar
  12. [12]
    Aldred, R. E. L., Širáň, J., Širáň, M.: A note on the number of graceful labelings of paths. Discrete Math., 261, 27–30 (2003)MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Burzio, M., Ferrarese, G.: The subdivision graph of a graceful tree is a graceful tree. Discrete Math., 181, 275–281 (1998)CrossRefMathSciNetGoogle Scholar
  14. [14]
    Rosa, A., Širáň, J.: Bipartite labelings of trees and the gracesize. J. Graph Theory, 19, 201–215 (1995)MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Snevily, H.: New families of graphs that have α-labelings. Discrete Math., 170, 185–194 (1997)MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Muntaner-Batle, F. A.: Ph.D. Thesis, Universitat Politécnica de Catalunya, Barcelona, 2001Google Scholar
  17. [17]
    Bača, M., Barrientos, C.: Graceful and edge-antimagic labelings. Ars Combin., in pressGoogle Scholar
  18. [18]
    Abrham, J., Kotzig, A.: Exponential lower bounds for the number of graceful numbering of snakes. Congressus Numerantium, 72, 163–174 (1990)MathSciNetGoogle Scholar
  19. [19]
    Barrientos, C.: Difference Vertex Labelings, Ph.D. Thesis, Universitat Politècnica de Catalunya, Barcelona, 2004Google Scholar
  20. [20]
    Muntaner-Batle, F. A., Rius-Font, M.: On the structure of path-like trees. Discussiones Math. Graph Theory, 28(2), 249–265 (2008)MATHMathSciNetGoogle Scholar
  21. [21]
    Acharya, B. D.: Elementary parallel transformations of graphs. AKCE International J. Graphs and Combinatorics, 1, 63–67 (2004)MATHMathSciNetGoogle Scholar
  22. [22]
    Hegde, S. M., Shetty, S.: On graceful trees. Appl. Math., E-Notes 2, 192–197 (2002)MATHMathSciNetGoogle Scholar
  23. [23]
    Bača, M., Lin, Y., Muntaner-Batle, F. A.: Normalized embeddings of path-like trees. Utilitas Math., 78, 11-31 (2009)Google Scholar
  24. [24]
    Bača, M., Lin, Y., Muntaner-Batle, F. A.: Edge-antimagic labelings of forest. Utilitas Math., in pressGoogle Scholar
  25. [25]
    Bača, M., Lin, Y., Muntaner-Batle, F. A.: Super edge-antimagic labelings of the path-like trees. Utilitas Math., 73 117–128 (2007)MATHGoogle Scholar
  26. [26]
    Ngurah, A. A. G., Baskoro, E. T., Simanjuntak, R.: On edge-magic total labeling of kC 4-snakes and path-like trees. MIHMI, in pressGoogle Scholar
  27. [27]
    Wang, Z., Yan, L., Zhang, Z.: Vertex distinguishing equitable total chromatic number of join graph. Acta Math. Appl. Sinica (English Series), 23(3), 433–438 (2007)MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    Zhai, M., Lü, C.: Path decomposition of graphs with given path length. Acta Math. Appl. Sinica (English Series), 22(4), 633–638 (2006)MATHCrossRefGoogle Scholar
  29. [29]
    Figueroa-Centeno, R. M., Ichishima, R., Muntaner-Batle, F. A., et al.: Labeling generating matrices. J. Combin. Math. Combin. Comput., 67, 189–216 (2008)MATHMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer Berlin Heidelberg 2009

Authors and Affiliations

  • Martin Bača
    • 1
  • Yu Qing Lin
    • 2
  • Francesc A. Muntaner-Batle
    • 3
  • Miquel Rius-Font
    • 4
  1. 1.Department of Applied MathematicsTechnical University in KošiceKošiceSlovakia
  2. 2.School of Electrical Engineering and Computer ScienceThe University of NewcastleNewcastleAustralia
  3. 3.Universitat Internacional de CatalunyaBarcelona, CatalunyaSpain
  4. 4.Departament de Matemàtica Aplicada i Telemàtica Universitat Politècnica de CatalunyaBarcelona, CatalunyaSpain

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