The Brush Number of the Two-Dimensional Torus

In this paper we are interested in the brush number of a graph - a concept introduced by McKeil and by Messinger, Nowakowski and Pralat. Our main aim in this paper is to determine the brush number of the two-dimensional torus. This answers a question of Bonato and Messinger. We also find the brush number of the cartesian product of a clique with a path, which is related to the Box Cleaning Conjecture of Bonato and Messinger.