The Middle Stair for Complete Bipartite Parallel Chip-Firing

Minkyu Jung

We prove the middle-stair conjecture for every complete bipartite graph. If a parallel chip-firing game on $K_{a,b}$ has configuration $\ddot{I}ƒ$ with $2ab-a-b<|\ddot{I}ƒ|<2ab$, then its eventu…