Parallel Chip Firing Game associated with n-cube orientations

TL;DR

This paper investigates the cycle structures in the parallel chip firing game on n-cube orientations, revealing the existence of cycles of various lengths and their specific properties.

Contribution

It characterizes the cycle lengths generated by parallel evolutions in the chip firing game on n-cubes, including the existence of even and odd length cycles.

Findings

Existence of even length cycles from 2 to 2^n

Existence of odd length cycles from 1 to 2^{n-1}-1, excluding 3

Cycle lengths depend on the parity and orientation of the n-cube

Abstract

We study the cycles generated by the chip firing game associated with n-cube orientations. We show the existence of the cycles generated by parallel evolutions of even lengths from 2 to $2^n$ on $H_n$ (n >= 1), and of odd lengths different from 3 and ranging from 1 to $2^{n-1}-1$ on $H_n$ (n >= 4).