Chip-firing Game Research Articles

The Tutte polynomial

This is a close approximation to the content of my lecture. After a brief survey of well known properties, I present some new interpretations relating to random graphs, lattice point enumeration, and chip firing games. I then examine complexity issues and concentrate in particular, on the existence of randomized approximation schemes. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 210–228, 1999

Chip-Firing and the Critical Group of a Graph

A variant of the chip-firing game on a graph is defined. It is shown that the set of configurations that are stable and recurrent for this game can be given the structure of an abelian group, and that the order of the group is equal to the tree number of the graph. In certain cases the game can be used to illuminate the structure of the group.

Chip firing and the tutte polynomial

It is shown that the generating function of critical configurations of a version of a chip firing game on a graphG is an evaluation of the Tutte polynomial ofG, thus proving a conjecture of Biggs [3].

Algebraic Potential Theory on Graphs

This paper encompasses a motley collection of ideas from several areas of mathematics, including, in no particular order, random walks, the Picard group, exchange rate networks, chip-firing games, cohomology, and the conductance of an electrical network. The linking threads are the discrete Laplacian on a graph and the solution of the associated Dirichlet problem. Thirty years ago, this subject was dismissed by many as a trivial specialisation of cohomology theory, but it has now been shown to have hidden depths. Plumbing these depths leads to new theoretical advances, many of which throw light on the diverse applications of the theory. 1991 Mathematics Subject Classification 05C50.

Universality of the chip-firing game

We prove that the parallel updating of the chip-firing game on undirected graphs is universal. To achieve that, we simulate any given two-register machine by chip configurations. As corollaries, we prove that for finite graphs there exists exponential transient time to reach periodic configurations as well as exponential cycles. Also, we prove, for infinite graphs, that the reachability problem is undecidable.

Chip-Firing Games on Mutating Graphs

We investigate a generalization of a chip-firing game on a graph of Björner, Lovász, and Shor [European J. Combin., 1 (1992), pp. 305–328]. In our version, the graph mutates during play. We show that some known results about the game length and period length of the earlier game hold for the mutating version as well, and some completely new bounds are also obtained. In a small detour, we treat an orientability concept for eulerian graphs.

The chip firing game onn-cycles

Spencer introduced a chip firing game on a line, which was generalized by Bjorner, Lovasz and Shor to graphs. This paper characterizes those configurations which may be repeated in a game, and analyzes the game forn-cycles.

No polynomial bound for the period of the parallel chip firing game on graphs

The following (solitaire) game is considered: Initially each node of a simple, connected, finite graph contains a finite number of chips. A move consists in firing all nodes with at least as many chips as their degree, where firing a node corresponds to sending one of the node's chips to each one of the node's neighbors.

Parallel chip firing games on graphs

We study the periodic behaviour of parallel dynamics associated with the chip firing game introduced by Spencer (1986). We prove that on trees the periods are of length one (fixed points) or two. For general graphs we give counter-examples where periods depend on the graph size. We also prove that the chip firing game may be used as an algorithm to evaluate logical functions, which shows that the dynamics have nontrivial computing capabilities. Finally, we study a related game on graphs.

No Polynomial Bound for the Chip Firing Game on Directed Graphs

Tardos has proved a polynomial bound on the length of a convergent chip firing game on an undirected graph. This paper demonstrates a game with exponential growth on a directed graph.

No polynomial bound for the chip firing game on directed graphs

Tardos has proved a polynomial bound on the length of a convergent chip firing game on an undirected graph. This paper demonstrates a game with exponential growth on a directed graph.

Polynomial Bound for a Chip Firing Game on Graphs

Bjorner, Lvasz, and Shor have introduced a chip firing game on graphs. This paper proves a polynomial bound on the length of the game in terms of the number of vertices of the graph provided the length is finite. The obtained bound is best possible within a constant factor.