Archiv der Mathematik

, Volume 95, Issue 1, pp 9–13 | Cite as

A constant bound for the periods of parallel chip-firing games with many chips

Article

Abstract

We prove that any parallel chip-firing game on a graph G with at least 4|E(G)| − |V(G)| chips stabilizes, i.e., such a game has eventual period of length 1. Furthermore, we obtain a polynomial bound on the number of rounds before stabilization. This result is a counterpoint to previous results which showed that the eventual periods of parallel chip-firing games with few chips need not be polynomially bounded.

Mathematics Subject Classification (2000)

05C35 05C85 68Q25 (Primary) 37B15 68R10 68Q80 (Secondary) 

Keywords

Chip-firing Parallel chip-firing Stabilization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Biggs N.L.: Chip-firing and the critical group of a graph. J. Algebraic Combin. 9, 25–45 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bitar J., Goles E.: Parallel chip firing games on graphs. Theoret. Comput. Sci. 92, 291–300 (1992)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Björner A., Lovász L., Shor P.: Chip-firing games on graphs. European J. Combin. 12, 283–291 (1991)MATHMathSciNetGoogle Scholar
  4. 4.
    Kiwi M.A. et al.: No polynomial bound for the period of the parallel chip firing game on graphs. Theoret. Comput. Sci. 136, 527–532 (1994)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kominers P.M.: The candy-passing game for c ≥ 3n − 2. Pi Mu Epsilon J. 12, 459–460 (2008)Google Scholar
  6. 6.
    López C.M.: Chip firing and the Tutte polynomial. Ann. Combin. 1, 253–259 (1997)MATHCrossRefGoogle Scholar
  7. 7.
    Spencer J.: Balancing vectors in the max norm. Combinatorica 6, 55–65 (1986)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Tardos G.: Polynomial bound for a chip firing game on graphs. SIAM J. Discr. Math. 1, 397–398 (1988)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of EconomicsMassachusetts Institute of TechnologyBethesdaUSA
  2. 2.Department of EconomicsHarvard University, and Harvard Business SchoolBostonUSA

Personalised recommendations