Toppling numbers of complete and random graphs

TL;DR

This paper analyzes a two-player chip-firing game on graphs, establishing asymptotic bounds for the toppling number of complete graphs and relating it to random graphs, revealing strategic and probabilistic insights.

Contribution

It introduces a game-theoretic approach to chip-firing on graphs, providing asymptotic bounds for complete graphs and linking the toppling number to random graph properties.

Findings

Bounds for the toppling number of complete graphs: approximately 0.5964 to 0.6372 times n^2.

Coupling of toppling numbers between complete and random graphs for certain edge probabilities.

Asymptotic equivalence of the toppling number in random graphs to a scaled version of the complete graph's toppling number.

Abstract

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 {\em toppling number} of a graph $G$, and is denoted by $\tg(G)$. 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 $n$ $$0.596400 n^2 < \tg(K_n) < 0.637152 n^2.$$ 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 $pn \ge n^{2/\sqrt{\log n}}$ asymptotically almost surely $t(G(n,p))=(1+o(1))p\tg(K_n)$.