Game Brush Number

TL;DR

This paper introduces a two-player game based on graph brushing processes, analyzing its behavior on complete and random graphs, and establishing asymptotic formulas for the game brush number.

Contribution

It models the game with differential equations and derives asymptotic values for the game brush number on complete and random graphs, connecting it to the original brush number.

Findings

The game brush number of the complete graph is approximately n^2/e.

For random graphs with p such that pn >> ln n, the game brush number is approximately p times that of the complete graph.

The study reveals relationships between the game brush number and the original brush number.

Abstract

We study a two-person game based on the well-studied brushing process on graphs. Players Min and Max alternately place brushes on the vertices of a graph. When a vertex accumulates at least as many brushes as its degree, it sends one brush to each neighbor and is removed from the graph; this may in turn induce the removal of other vertices. The game ends once all vertices have been removed. Min seeks to minimize the number of brushes played during the game, while Max seeks to maximize it. When both players play optimally, the length of the game is the game brush number of the graph $G$, denoted $b_g(G)$. By considering strategies for both players and modelling the evolution of the game with differential equations, we provide an asymptotic value for the game brush number of the complete graph; namely, we show that $b_g(K_n) = (1+o(1))n^2/e$. Using a fractional version of the game, we couple the game brush numbers of complete graphs and the binomial random graph $\mathcal{G}(n,p)$. It is shown that for $pn \gg \ln n$ asymptotically almost surely $b_g(\mathcal{G}(n,p)) = (1 + o(1))p b_g(K_n) = (1 + o(1))pn^2/e$. Finally, we study the relationship between the game brush number and the (original) brush number.