Related papers: Parallel Chip Firing Game associated with n-cube o…
The parallel chip-firing game is a periodic automaton on graphs in which vertices "fire" chips to their neighbors. In 1989, Bitar conjectured that the period of a parallel chip-firing game with n vertices is at most n. Though this…
We introduce a natural variant of the parallel chip-firing game, called the diffusion game. Chips are initially assigned to vertices of a graph. At every step, all vertices simultaneously send one chip to each neighbour with fewer chips. As…
The parallel chip-firing game is an automaton on graphs in which vertices "fire" chips to their neighbors when they have enough chips to do so. The game is always periodic, and we concern ourselves with the firing sequences of vertices. We…
In 1992, Bitar and Goles introduced the parallel chip-firing game on undirected graphs. Two years later, Prisner extended the game to directed graphs. While the properties of parallel chip-firing games on undirected graphs have been…
Parallel Diffusion is a variant of Chip-Firing introduced in 2018 by Duffy et al. In Parallel Diffusion, chips move from places of high concentration to places of low concentration through a discrete-time process. At each time step, every…
We continue our studies of burn-off chip-firing games from [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121-132; MR3040546] and [Australas. J. Combin. 68 (2017), no. 3, 330-345; MR3656659]. The latter article introduced randomness…
Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices. We study chip-firing on an infinite binary tree in which we add a self-loop to the root to ensure each vertex has degree 3. A vertex can…
The enhanced hypercube $Q_{n,k}$ is a variant of the hypercube $Q_n$. We investigate all the lengths of cycles that an edge of the enhanced hypercube lies on. It is proved that every edge of $Q_{n,k}$ lies on a cycle of every even length…
We study how parallel chip-firing on the complete graph K_n changes behavior as we vary the total number of chips. Surprisingly, the activity of the system, defined as the average number of firings per time step, does not increase smoothly…
If $n$ is even, the $n$-dimensional hypercube can be decomposed into edge-disjoint cycles of length $2^i$ for every value of $i$ from $2$ to $n$.
In 2010, Kominers and Kominers proved that any parallel chip-firing game on $G(V,\,E)$ with $|\sigma|\geq 4|E|-|V|$ chips stabilizes. Recently, Bu, Choi, and Xu made the bound exact: all games with $|\sigma|< |E|$ chips or $|\sigma|>…
We consider edge decompositions of the $n$-dimensional hypercube $Q_n$ into isomorphic copies of a given graph $H$. While a number of results are known about decomposing $Q_n$ into graphs from various classes, the simplest cases of paths…
We study labeled chip-firing on binary trees and some of its modifications. We prove a sorting property of terminal configurations of the process. We also analyze the endgame moves poset and prove that this poset is a modular lattice.
We study chip-firing games on multigraphs whose underlying simple graphs are trees, paths, and stars, denoted as banana trees, paths, and stars respectively. We present a polynomial time algorithm to compute the divisorial gonality of…
We study a two-person game played on graphs based on the widely studied chip-firing game. Players Max and Min alternately place chips on the vertices of a graph. When a vertex accumulates as many chips as its degree, it fires, sending one…
The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional…
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…
It is known that the $n$-dimensional hypercube $Q_n,$ for $n$ even, has a decomposition into $k$-cycles for $k=n, 2n,$ $2^l$ with $2 \leq l \leq n.$ In this paper, we prove that $Q_n$ has a decomposition into $2^mn$-cycles for $n \geq 2^m.$…
We introduce parallelism into the basic algebra of games to model concurrent game algebraically. Parallelism is treated as a new kind of game operation. The resulted algebra of concurrent games can be used widely to reason the parallel…
The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional twisted cube, denoted by $TQ_n$, an important variation of the hypercube, possesses some…