Toppling numbers of complete and random graphs

Anthony Bonato,William B Kinnersley,Pawel Pralat

doi:10.46298/dmtcs.2084

Abstract

Graph Theory 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 chip to each neighbour; this may in turn cause other vertices to fire. The game ends when vertices continue firing forever. Min seeks to minimize the number of chips played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the toppling number of a graph G, and is denoted by t(G). By considering strategies for both players and investigating the evolution of the game with differential equations, we provide asymptotic bounds on the toppling number of the complete graph. In particular, we prove that for sufficiently large n 0.596400 n2 < t(Kn) < 0.637152 n2. Using a fractional version of the game, we couple the toppling numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≥n² / √ log(n) asymptotically almost surely t(G(n,p))=(1+o(1)) p t(Kn).

Highlights

We study a two-person game played on graphs based on the widely studied chip-firing game
The game of chip-firing and its variants have been a subject of active investigation in a variety of disciplines, with applications to topics such as Tutte polynomials, spectral graph theory, matroids, and statistical mechanics; see [11] for a survey with an extensive bibliography
The Abelian Sandpile model is notable as it is a dynamical system that displays self-organized criticality

Summary

Introduction

The game of chip-firing and its variants have been a subject of active investigation in a variety of disciplines, with applications to topics such as Tutte polynomials, spectral graph theory, matroids, and statistical mechanics; see [11] for a survey with an extensive bibliography. We consider the following game-theoretic synthesis of the chip-firing game and the Abelian Sandpile model played on undirected (finite) graphs. Lovasz, and Shor [3] proved that for any configuration of any graph, the order of vertex firings does not matter. If c is a volatile configuration, after any list of vertex firings, the resulting configuration remains volatile; if instead c is not volatile, any two maximal lists of vertex firings yield the same stable configuration They showed that every volatile configuration of a graph G has at least |E(G)| chips, and that every configuration having at least 2 |E(G)| − |V (G)| + 1 chips is volatile. We may end up firing vertices in a different order than if we had always fired immediately; by the result of Bjorner, Lovasz, and Shor, this does not affect whether or not the current configuration is volatile. For background on graph theory, the reader is directed to [15]

Main results

Complete Graphs

The “triangle strategy” for Min

Random Graphs