On the limited increment parallel chip-firing game

In the same paper where Bitar and Goles defined the parallel chip-firing game, they also defined the limited increment parallel chip-firing game to be a chip-firing game on a graph G(V,E) in which each vertex fires as in the parallel chip-firing game but can receive at most one chip in each round. While the parallel chip-firing game has been widely-studied, the limited increment parallel chip-firing game has not been. We hereby prove the analogs of the core results of the parallel chip-firing game for the limited increment parallel chip-firing game: we exactly determine the possible cycle lengths for special classes of G(V,E), show that the cycle length cannot be bounded by a polynomial in the number of chips or vertices, and determine the possible atomic firing sequences.