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

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.

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Correspondence to Scott Duke Kominers.

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The second author gratefully acknowledges the support of a Harvard Mathematics Department Highbridge Fellowship.

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Kominers, P.M., Kominers, S.D. A constant bound for the periods of parallel chip-firing games with many chips. Arch. Math. 95, 9–13 (2010). https://doi.org/10.1007/s00013-010-0129-x

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Mathematics Subject Classification (2000)

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

Keywords

  • Chip-firing
  • Parallel chip-firing
  • Stabilization