Game Brush Number

Authors

William B. Kinnersley, University of Rhode IslandFollow
Pawel Pralat

Document Type

Article

Date of Original Version

2016

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 modeling 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))n²/e. Using a fractional version of the game, we couple the game brush numbers of complete graphs and the binomial random graph G(n,p). It is shown that for pn ≫ ln n asymptotically almost surely b_g(G(n,p))=(1+o(1))pb_g(K_n)=(1+o(1))pn²/e. Finally, we study the relationship between the game brush number and the (original) brush number.

Citation/Publisher Attribution

Kinnersley, W. B. & Pralat, P. (2016). Game Brush Number. Discrete Applied Mathematics, 207(10), 1-14. doi: 10.1016/j.dam.2016.02.011

Available at: http://dx.doi.org/10.1016/j.dam.2016.02.011