Uniform Bounds for Non-negativity of the Diffusion Game

TL;DR

This paper proves that for the diffusion game on any n-vertex graph, having at least n-2 chips per vertex guarantees non-negativity throughout the process, establishing a tight uniform bound.

Contribution

The paper establishes the optimal uniform bound of n-2 chips per vertex to prevent negative labels in the diffusion game, answering a question posed by Long and Narayanan.

Findings

The bound f(n)=n-2 is sufficient for non-negativity.

This bound is proven to be the best possible.

Similar bounds g(d) are considered for graphs with maximum degree d.

Abstract

We study a variant of the chip-firing game called the diffusion game. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and then for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. Long and Narayanan asked whether there exists an $f(n)$ for each $n$, such that whenever we have a graph on $n$ vertices and an initial allocation with at least $f(n)$ chips on each vertex, then the number of chips on each vertex will remain non-negative. We answer their question in the affirmative, showing further that $f(n)=n-2$ is the best possible bound. We also consider the existence of a similar bound $g(d)$ for each $d$, where $d$ is the maximum degree of the graph.