Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Computer Science Logic

CSL 2008: Computer Science Logic pp 339–353Cite as

On Counting Generalized Colorings

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5213))

Abstract

It is well known that the number of proper k-colorings of a graph is a polynomial in k. We investigate in this talk under what conditions a numeric graph invariant which is parametrized with parameters k 1, ..., k m is a polynomial in these parameters. We give a sufficient conditions for this to happen which is general enough to encompass all the graph polynomials which are definable in Second Order Logic. This not only covers the various generalizations of the Tutte polynomials, Interlace polynomials, Matching polynomials, but allows us to identify new graph polynomials related to combinatorial problems discussed in the literature.

This is a preview of subscription content, 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   109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   139.00
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alon, N., Ding, G., Oporowski, B., Vertigan, D.: Partitioning into graphs with only small components. Journal of Combinatorial Theory, Series B 87, 231–243 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  3. Bollobás, B.: Modern Graph Theory. Springer, Heidelberg (1999)

    Google Scholar 

  4. Courcelle, B., Godlin, B., Makowsky, J.A.: Towards a theory of graph polynomials, I: Second order definable polynomials (in preparation, 2007)

    Google Scholar 

  5. Courcelle, B.: A multivariate interlace polynomial (December 2006) (preprint)

    Google Scholar 

  6. Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer, Heidelberg (1996)

    MATH  Google Scholar 

  7. Dong, F.M., Koh, K.M., Teo, K.L.: Chromatic Polynomials and Chromaticity of Graphs. World Scientific, Singapore (2005)

    MATH  Google Scholar 

  8. Edwards, K.: The harmonious chromatic number and the achromatic number. In: Bailey, R.A. (ed.) Survey in Combinatorics. London Math. Soc. Lecture Note Ser, vol. 241, pp. 13–47. Cambridge Univ. Press, Cambridge (1997)

    Google Scholar 

  9. Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic. Springer, Heidelberg (1995)

    MATH  Google Scholar 

  10. Ebbinghaus, H.D., Flum, J., Thomas, W.: Mathematical Logic, 2nd edn. Undergraduate Texts in Mathematics. Springer, Heidelberg (1994)

    MATH  Google Scholar 

  11. Edwards, K., McDiarmid, C.: The complexity of harmonious colouring for trees. Discrete Appl. Math. 57(2-3), 133–144 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall, Boca Raton (1993)

    MATH  Google Scholar 

  13. Harary, F., Hedetniemi, S., Rins, G.: An interpolation theorem for graphical homomorphisms. Portugal. Math. 26, 453–462 (1967)

    MATH  MathSciNet  Google Scholar 

  14. Hopcroft, J.E., Krishnamoorthy, M.S.: On the harmonious coloring of graphs. SIAM J. Algebraic Discrete Methods 4, 306–311 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hughes, F., MacGillivray, G.: The achromatic number of graphs: a survey and some new results. Bull. Inst. Combin. Appl. 19, 27–56 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  17. Linial, N., Matoušek, J., Sheffet, O., Tardos, G.: Graph coloring with no large monochromatic components (2007) arXiv:math/0703362v1

    Google Scholar 

  18. Makowsky, J.A.: Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic 126(1-3), 159–213 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Makowsky, J.A.: From a zoo to a zoology: Descriptive complexity for graph polynomials. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 330–341. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  20. Makowsky, J.A., Zilber, B.: Polynomial invariants of graphs and totally categorical theories. MODNET Preprint No. 21 (2006), http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server

  21. Noble, S.D., Welsh, D.J.A.: A weighted graph polynomial from chromatic invariants of knots. Ann. Inst. Fourier, Grenoble 49, 1057–1087 (1999)

    MATH  MathSciNet  Google Scholar 

  22. Sokal, A.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Survey in Combinatorics, 2005. London Mathematical Society Lecture Notes, vol. 327, pp. 173–226 (2005)

    Google Scholar 

  23. Toda, S., Watanabe, O.: Polynomial time 1-Turing reductions from #PH to #P. Theor. Comp. Sc. 100, 205–221 (1992)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Technion-Israel Institute of Technology, Haifa, Israel

    T. Kotek & J. A. Makowsky

  2. Mathematical Institute, University of Oxford,  

    B. Zilber

Authors
  1. T. Kotek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Makowsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Zilber
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Michael Kaminski Simone Martini

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kotek, T., Makowsky, J.A., Zilber, B. (2008). On Counting Generalized Colorings. In: Kaminski, M., Martini, S. (eds) Computer Science Logic. CSL 2008. Lecture Notes in Computer Science, vol 5213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87531-4_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-87531-4_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-87530-7

  • Online ISBN: 978-3-540-87531-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation