Related papers: Chip-firing game and partial Tutte polynomial for …

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…

We give a new characterization of the Tutte polynomial of graphs. Our characterization is formally close (but inequivalent) to the original definition given by Tutte as the generating function of spanning trees counted according to…

For integers \(r\ge 2\), \(t\ge 1\) and a real number \(a\in(3/2,2]\), we study the typical structure of oriented graphs and digraphs that do not contain a blow-up \(T_{r+1}^t\) of a transitive tournament. We prove that almost every…

In the chip-firing variant, Diffusion, chips flow from places of high concentration to places of low concentration (or equivalently, from the rich to the poor). We explore this model on complete graphs, determining the number of different…

In this paper, we study the complexity of the chip-firing reachability problem. We show that for Eulerian digraphs, the reachability problem can be decided in strongly polynomial time, even if the digraph has multiple edges. We also show a…

We study the interplay between chip-firing games and potential theory on graphs, characterizing reduced divisors ($G$-parking functions) on graphs as the solution to an energy (or potential) minimization problem and providing an algorithm…

For any finite partially ordered set $P$, the $P$-Eulerian polynomial is the generating function for the descent number over the set of linear extensions of $P$, and is closely related to the order polynomial of $P$ arising in the theory of…

In this paper, we introduce a two-player impartial game on graphs, called a {\em feedback game}, which is a variant of the generalized geography. We study the feedback game on Eulerian graphs. In particular, we show that the…

Chip-firing on a directed graph is a game in which chips, a discrete commodity, are placed on the vertices of the graph and are transferred between vertices. In this paper, we study a chip-firing game on the Hasse diagram of the lattice of…

The semi-random graph process is a single-player game that begins with an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex…

We investigate a cancellation property satisfied by a connected Eulerian digraph $D$. Namely, unless $D$ is a single directed cycle, we have $\sum_{k\geq 1} (-1)^{k} f_k(D)=0$, where $f_k(D)$ is the number of partitions of Eulerian circuits…

A directed graph is called Eulerian, if it contains a tour that traverses every arc in the graph exactly once. We study the problem of Eulerian extension (EE) where a directed multigraph G and a weight function is given and it is asked…

We propose a generalization of the graphical chip-firing model allowing for the redistribution dynamics to be governed by any invertible integer matrix while maintaining the long term critical, superstable, and energy minimizing behavior of…

In this article, we provide three formulas allowing to compute the minimum amount of initial chips leading to an infinite Chip-firing game. These formulas hold for strongly connected directed loop-free multigraphs and generalize what was…

In [5], Holroyd, Levine, M\'esz\'aros, Peres, Propp and Wilson characterize recurrent chip-and-rotor configurations for strongly connected digraphs. However, the number of steps needed to recur, and the number of orbits is left open for…

Tutte's celebrated barycentric embedding theorem describes a natural way to build straight-line embeddings (crossing-free drawings) of a (3-connected) planar graph: map the vertices of the outer face to the vertices of a convex polygon, and…

We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on…

We prove several results concerning a polynomial that arises from the sandpile model on directed graphs; these results are previously only known for undirected graphs. Implicit in the sandpile model is the choice of a sink vertex, and it is…

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…

This paper considers a natural ruleset for playing a partisan combinatorial game on a directed graph, which we call Digraph Placement. Given a digraph $G$ with a not necessarily proper $2$-coloring of $V(G)$, the Digraph Placement game…