On lengths of burn-off chip-firing games

P. Mark Kayll, Dave Perkins1

Research output: Contribution to journalArticlepeer-review


We continue our studies of burn-off chip-firing games from [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121-132; MR3040546] and [Australas. J. Combin. 68 (2017), no. 3, 330-345; MR3656659]. The latter article introduced randomness by choosing successive seeds uniformly from the vertex set of a graph G. The length of a game is the number of vertices that fire (by sending a chip to each neighbor and annihilating one chip) as an excited chip configuration passes to a relaxed state. This article determines the probability distribution of the game length in a long sequence of burn-off games. Our main results give exact counts for the number of pairs (C, v), with C a relaxed legal configuration and v a seed, corresponding to each possible length. In support, we give our own proof of the well-known equicar-dinality of the set R of relaxed legal configurations on G and the set of spanning trees in the cone G∗ of G. We present an algorithmic, bijective proof of this correspondence.

Original languageEnglish
Pages (from-to)53-77
Number of pages25
JournalJournal of Combinatorial Mathematics and Combinatorial Computing
StatePublished - Feb 2021


  • Burn-off game
  • Chip-firing
  • Game-length probability
  • Markov chain
  • Relaxed legal configuration
  • Sandpile group
  • Spanning tree


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