# Chip-firing game

Example graph
A possible firing sequence

The chip-firing game is a one-player game on a graph which was invented around 1983 and since has become an important part of the study of structural combinatorics. The set of configurations that are stable and recurrent for this game can be given the structure of an abelian group. In addition, the order of the group is equal to the tree number of the graph.[1][2]

## Definition

Let graph G be a finite, simple, connected graph without loops such that G = (V, E) where V = {1, 2,..., n} and E = {e1, e2,..., em}. Let deg(vi) be the degree of vertex i where i = {v ∈ V(G)}. Each vertex of the graph will be assigned a nonnegative integer S(vi). Let S be the chip configurations of the game. S will show us how many chips are available at vi. A move will consist of selecting a vertex vj which has at least as many chips on it, as its degree; thus S(v) ≥ deg(v). One chip will be fired along each of its incident edges. Each time vj is fired, it loses its degree in chips. (i.e. If the deg(v) = 2 when v is fired, v loses two chips.) Each of vj neighbouring vertices will gain one chip per edge incident with vj. Let v(v, w) be the number of edges joining vj to w. Let the value of x(v) be the number of times v is fired in the sequence of moves.

After the sequence of firing s, the new chip configurations, s′ is given by:

${\displaystyle s'(v)=s(v)-x(v)\deg(v)+\sum _{v\neq w}x(w)v(v,w)}$
• If the number of chips is less than the number of edges, the game is always finite.
• If the number of chips is at least the number of edges the game can be infinite for an appropriately chosen initial configuration.
• If the number of chips is more than twice the number of edges minus the number of vertices, then the game is always infinite.
• The game terminates if each vertex has fewer chips than its degree.

## Variants

In Biggs's variant of the chip fire game (also known as dollar game), one vertex q is allowed to go into debt. In other words, it is allowed to take a negative integer value, unlike all the other vertices. This vertex q is called the "bank". The bank does not fire. It just sits there collecting chips. Eventually, q will accumulate so many chips that no other vertex can fire which will make the configuration stable. Once the configuration is stable, vertex q can fire chips to neighbouring vertices to jump start the 'economy'. The bank can only fire if and only if the current configuration is stable.

Thus, in this alternate game, a configuration s is an integer-valued function satisfying:

${\displaystyle s(q)=-\sum s(v)\leq 0{\text{ where }}s(v)\geq 0{\text{ and }}\neq q.}$

We define a stable chip configuration to be

${\displaystyle 0\leq s(v)<\deg(g)\qquad (v\neq q).}$