Skip to main content
SpringerLink
Log in

The Polynomial Method

  • Chapter
  • First Online:
Book cover Graceful, Harmonious and Magic Type Labelings

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

  • 566 Accesses

Abstract

In Chap. 6 we discussed the existence of labelings by utilizing the ⊗ h -product of digraphs, which could be expressed algebraically as a generalization of voltage assignments, a classical technique used in topological graph theory. In this chapter, we introduce an algebraic method: Combinatorial Nullstellensatz.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

References

  1. Alon, N.: Combinatorial Nullstellensatz. Comb. Probab. Comput. 8, 7–29 (1999)

    Article  Google Scholar 

  2. Alon, N., Tarsi, M.: Colorings and orientations of graphs. Combinatorica 12, 125–134 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Càmara, M., Lladó, A., Moragas, J.: On a conjecture of Graham and Häggkvist with the polynomial method. Eur. J. Comb. 34, 1585–1592 (2013)

    MATH  Google Scholar 

  4. Dyson, F.J.: Statistical theory of the energy levels of complex systems I. J. Math. Phys. 3, 140–156 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gunson, J.: Proof of a conjecture of Dyson in the statistical theory of energy levels. J. Math. Phys. 3, 752–753 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  6. Häggkvist, R.L.: Decompositions of complete bipartite graphs. In: Surveys in Combinatorics, pp. 115–146. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  7. Hartsfield, N., Ringel, G.: Pearls in Graph Theory. A Comprehensive Introduction. Academic, Boston (1990)

    MATH  Google Scholar 

  8. Hefetz, D.: Anti-magic graphs via the Combinatorial Nullstellensatz. J. Graph Theory 50, 263–272 (2005). http://dx.doi.org/10.1002/jgt.20112

    Article  MathSciNet  MATH  Google Scholar 

  9. Kézdy, A.E.: ρ-valuations for some stunted trees. Discrete Math. 306, 2786–2789 (2006). http://dx.doi.org/10.1016/j.disc.2006.05.026

  10. Kotzig, A.: On certain vertex-valuations of finite graphs. Utilitas Math. 4, 261–290 (1973)

    MathSciNet  MATH  Google Scholar 

  11. Lladó, A., López, S.C.: Edge-decompositions of K n, n into isomorphic copies of a given tree. J. Graph Theory 48, 1–18 (2005). http://dx.doi.org/10.1002/jgt.20024

  12. Lladó, A., López, S.C., Moragas, J.: Every tree is a large subtree of a tree that decomposes K n or K n, n . Discret. Math. 310, 838–842 (2010). http://dx.doi.org/10.1016/j.disc.2009.09.021

  13. Lladó, A., Ringel, G., Serra, O.: Decomposition of complete bipartite graphs into trees. Tech. Rep. DMAT Research Report 11/96, Universitat Politècnica de Catalunya (1996)

    Google Scholar 

  14. Lo, S.: On edge-graceful labelings of graphs. Congr. Numer. 50, 231–241 (1985)

    MathSciNet  MATH  Google Scholar 

  15. Rosa, A.: On certain valuations of the vertices of a graph. In: Theory of Graphs (Internat. Symposium, Rome, July 1966), pp. 349–355. Gordon and Breach, New York; Dunod, Paris (1967)

    Google Scholar 

  16. Wilson, K.: Proof of a conjecture of Dyson. J. Math. Phys. 3, 1040–1043 (1962)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The proofs from [9] are introduced with permission from [9], Elsevier, ©2006. The proofs from [12] are introduced with permission from [12], Elsevier, ©2009.

Author information

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

Authors
  1. Susana C. López
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Francesc A. Muntaner-Batle
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 The Author(s)

About this chapter

Cite this chapter

López, S.C., Muntaner-Batle, F.A. (2017). The Polynomial Method. In: Graceful, Harmonious and Magic Type Labelings. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-52657-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation