The Middle Stair for Complete Bipartite Parallel Chip-Firing

Abstract

We prove the middle-stair conjecture for every complete bipartite graph. If a parallel chip-firing game on $K_{a,b}$ has configuration $σ$ with $2ab-a-b<|σ|<2ab$, then its eventual period is $2$. The balanced case $K_{a,a}$ was proved by Ji, Li, and Wang using one-parameter conjugate configurations. We introduce two-parameter conjugates $c^{k,\ell}$, in which the rank shift on one side supplies the additive offset on the other. These conjugates preserve both the total number of chips and the activity. An exact Ferrers-diagram count then produces a nonnegative conjugate with two-round firing coverage on one side. The coverage propagates in alternating two-round waves, giving activity $1/2$; non-clumpiness then forces period $2$.