An Algebraic Characterization of Rainbow Connectivity

  • Prabhanjan Ananth
  • Ambedkar Dukkipati
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)


The use of algebraic techniques to solve combinatorial problems is studied in this paper. We formulate the rainbow connectivity problem as a system of polynomial equations. We first consider the case of two colors for which the problem is known to be hard and we then extend the approach to the general case. We also present a formulation of the rainbow connectivity problem as an ideal membership problem.


Graphs NulLA alogirithm ideal membership 


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  1. 1.
    Alon, N.: Combinatorial Nullstellensatz. Combinatorics, Probability and Computing 8(1&2), 7–29 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Lovász, L.: Stable sets and polynomials. Discrete Mathematics 124(1-3), 137–153 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    De Loera, J.: Gröbner bases and graph colorings. Beiträge Algebra Geom. 36(1), 89–96 (1995)zbMATHGoogle Scholar
  4. 4.
    De Loera, J., Lee, J., Malkin, P., Margulies, S.: Hilbert’s Nullstellensatz and an algorithm for proving combinatorial infeasibility. In: ISSAC 2008: Proceedings of the Twenty-first International Symposium on Symbolic and Algebraic Computation, pp. 197–206. ACM (2008)Google Scholar
  5. 5.
    Margulies, S.: Computer algebra, combinatorics, and complexity: Hilberts Nullstellensatz and NP-complete problems. PhD thesis, University of California (2008)Google Scholar
  6. 6.
    Bayer, D.: The division algorithm and the Hilbert scheme. PhD thesis, Harvard University (1982)Google Scholar
  7. 7.
    Cox, D.A., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Undergraduate Texts in Mathematics. Springer (2007)Google Scholar
  8. 8.
    Kollár, J.: Sharp effective Nullstellensatz. American Mathematical Society 1(4) (1988)Google Scholar
  9. 9.
    Brownawell, W.: Bounds for the degrees in the Nullstellensatz. The Annals of Mathematics 126(3), 577–591 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Loera, J., Lee, J., Margulies, S., Onn, S.: Expressing combinatorial problems by systems of polynomial equations and hilberts nullstellensatz. Combinatorics, Probability and Computing 18(04), 551–582 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chartrand, G., Johns, G., McKeon, K., Zhang, P.: Rainbow connection in graphs. Math. Bohem 133(1), 85–98 (2008)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Alon, N., Tarsi, M.: A note on graph colorings and graph polynomials. Journal of Combinatorial Theory Series B 70, 197–201 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Prabhanjan Ananth
    • 1
  • Ambedkar Dukkipati
    • 1
  1. 1.Department of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia

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