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Graceful Labelings: The Shifting Technique

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Graceful, Harmonious and Magic Type Labelings

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Abstract

Graceful labelings of graphs appeared in 1967 due to the relationship found with the problem of decompositions of graphs, in particular with the problem of decomposing complete graphs into copies of a given tree. Strong relations between graceful labelings and Golomb rulers (which are a different way to understand Sidon sets) were also found.

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References

  1. Acharya, B.D.: Construction of certain infinite families of graceful graphs from a given graceful graph. Def. Sci. J. 32 (3), 231–236 (1982)

    Article  MATH  Google Scholar 

  2. Adamaszek, M.: Efficient enumeration of graceful permutations. J. Combin. Math. Combin. Comput. 87, 191–197 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Aldred, R.E.L., Mckay, B.D.: Graceful and harmonius labelings of trees. Bull. Inst. Combin. Appl. 23, 69–72 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Aldred, R.E.L., Siran, J., Siran, M.: A note on the number of graceful labelings of paths. Discrete Math. 261, 27–30 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aravamudhan, R., Murugan, M.: Numbering of the vertices of K a, 1, b (preprint)

    Google Scholar 

  6. Arumugam, S., Bagga, J.: Graceful labeling algorithms and complexity - a survey. J. Indones. Math. Soc, 1–9 (2011). Special edn.

    Google Scholar 

  7. Babcock, W.C.: Intermodulation interference in radio Systems/Frequency of occurrence and control by channel selection. Bell Syst. Tech. J. 31, 63–73 (1953)

    Article  Google Scholar 

  8. Bača, M., Lin, Y., Muntaner-Batle, F.A.: Super edge-antimagic labelings of the path-like trees. Util. Math. 73, 117–128 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Bagga, J., Heinz, A., Majumder, M.M.: An algorithm for computing all graceful labelings of cycles. Congr. Numer. 186, 57–63 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Balbuena, C., García-Vázquez, P., Marcote, X., Valenzuela, J.C.: Trees having an even or quasi even degree sequence are graceful. Appl. Math. Lett. 20, 370–375 (2007). http://dx.doi.org/10.1016/j.aml.2006.04.020

    Article  MathSciNet  MATH  Google Scholar 

  11. Barrientos, C.: Difference vertex labelings. Ph.D. thesis, Universitat Politècnica de Catalunya (2004)

    Google Scholar 

  12. Bermond, J.C.: Graceful graphs, radio antennae, and French windmills. In: Graph Theory and Combinatorics, pp. 18–37. Pitman Publishing Ltd., London (1979)

    Google Scholar 

  13. Bloom, G.S.: A counterexample to a theorem of S. Piccard. J. Combin. Theory (A) 22, 378–379 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bodendiek, R., Schumacher, H., Wegner, H.: Uber graziose graphen. Math. Phys. Semesterberichte 24, 103–106 (1977)

    MATH  Google Scholar 

  15. Burzio, M., Ferrarese, G.: The subdivision graph of a graceful tree is a graceful tree. Discrete Math. 181, 275–281 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Delorme, C., Maheo, M., Thuillier, H., Koh, K.M., Teo, H.K.: Cycles with a chord are graceful. J. Graph Theory 4, 409–415 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. El-Zanati, S.I., Kenig, M.J., Eynden, C.V.: Near α-labelings of bipartite graphs. Aust. J. Combin. 21, 275–285 (2000)

    MathSciNet  MATH  Google Scholar 

  18. Fang, W.: A computational approach to the graceful tree conjecture (2010). http://arxiv.org/abs/1003.3045

    Google Scholar 

  19. Gallian, J.A.: Living with the labeling disease for 25 years. J. Indones. Math. Soc. 54–88 (2011). Special edn.

    Google Scholar 

  20. Gallian, J.A.: A dynamic survey of graph labeling. Electron. J. Combin. 19 (DS6) (2016)

    Google Scholar 

  21. Gardner, M.: Mathematical games: the graceful graphs of Solomon Golomb, or how to number a graph parsimoniously. Sci. Am. 226 (3/4/6), 108–112; 104; 118 (1972)

    Google Scholar 

  22. Gnanajothi, R.B.: Topics in Graph Theory. Ph.D. thesis, Madurai Kamaraj University (1991)

    Google Scholar 

  23. Golomb, S.W.: How to number a graph. In: Graph Theory and Computing, pp. 23–37. Academic, New York (1972)

    Google Scholar 

  24. Grannell, M., Griggs, T., Holroy, F.: Modular gracious labelings of trees. Discrete Math. 231, 199–219 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hegde, S., Shetty, S.: On graceful trees. Appl. Math. E-Notes 2, 192–197 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Horton, M.: Graceful trees: statistics and algorithms. Ph.D. thesis, University of Tasmania (2003)

    Google Scholar 

  27. Hrnčiar, P., Haviar, A.: All trees of diameter five are graceful. Discrete Math. 233, 133–150 (2001). http://dx.doi.org/10.1016/S0012-365X(00)00233-8

    Article  MathSciNet  MATH  Google Scholar 

  28. IBM: http://www.research.ibm.com/people/s/shearer/grtab.html. Accessed June (2016)

  29. Koh, K.M., Rogers, D.G., Tan, T.: On graceful trees. Nanta Math. 10 (2), 207–211 (1997)

    MathSciNet  MATH  Google Scholar 

  30. López, S.C., Muntaner-Batle, F.A.: A new application of the ⊗ h -product to α-labelings. Discrete Math. 338, 839–843 (2015)

    Article  MathSciNet  Google Scholar 

  31. Mavronicolas, M., Michael, L.: A substitution theorem for graceful trees and its applications. Discrete Math. 309, 3757–3766 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mixton, W.: cited in [21]

    Google Scholar 

  33. Muntaner-Batle, F.A., Rius-Font, M.: On the structure of path-like trees. Discuss. Math. Graph Theory 28, 249–265 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Piccard, S.: Sur les ensembles de distances des ensembles de points d’un espace euclidien (1939)

    MATH  Google Scholar 

  35. Ringel, G.: Problem 25. In: Theory of Graphs and its Application (Proceedings of Symposium, Smolenice 1963), p. 162. Nakl. CSAV, Praha (1964)

    Google Scholar 

  36. Robinson, J.P., Bernstein, A.J.: A class of binary recurrent codes with limited error propagation. IEEE Trans. Inf. Theory IT-13, 106–113 (1967)

    Article  MATH  Google Scholar 

  37. Rosa, A.: On certain valuations of the vertices of a graph. In: Gordon, N. Breach, D. Paris (eds.) Theory of Graphs (International Symposium, Rome, July 1966), pp. 349–355. Gordon and Breach/Dunod, New York/Paris (1967)

    Google Scholar 

  38. Sidon, S.: Ein statz über trigonometrishe polynome und seine anwendung in der thorie der Fourier Reihen. Math. Ann. 106, 536–539 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  39. Simmons, G.J.: Synch-sets: a variant of difference sets. Congr. Numer. 10, 625–645 (1974)

    MathSciNet  MATH  Google Scholar 

  40. Snevily, H.S.: New families of graphs that have α-labelings. Discrete Math. 170, 185–194 (1997). http://dx.doi.org/10.1016/0012-365X(95)00159-T

    Article  MathSciNet  MATH  Google Scholar 

  41. Stanton, R., Zarnke, C.: Labeling of balanced trees. In: Proceedings of 4th Southeast Conference on Combinatorics Graph Theory and Computing, pp. 479–495. Utilitas Mathematica, Boca Raton (1973)

    Google Scholar 

  42. Truszczyński, M.: Graceful unicyclic graphs. Demonstatio Math. 17, 377–387 (1984)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The proofs from [10] and Fig. 5.9 are introduced with permission from [10], Elsevier, ©2006. The proofs from [27] are introduced with permission from [27], Elsevier, ©2001. We gratefully acknowledge permission to use [37] by its author. We also gratefully acknowledge permission to use [39] by the publisher of Congr. Numer. The proof from [40] is introduced with permission from [40], Elsevier, ©1997.

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Authors and Affiliations

  1. Department of Mathematics, Universitat Politècnica de Catalunya, Castelldefels, Spain

    Susana C. López

  2. School of Electrical Engineering and Computer Science Faculty of Engineering and Built Environment, The University of Newcastle, Newcastle, NSW, Australia

    Francesc A. Muntaner-Batle

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  1. Susana C. López
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  2. Francesc A. Muntaner-Batle
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López, S.C., Muntaner-Batle, F.A. (2017). Graceful Labelings: The Shifting Technique. In: Graceful, Harmonious and Magic Type Labelings. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-52657-7_5

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