Brushing without capacity restrictions

In graph cleaning problems, brushes clean a graph by traversing it subject to certain rules. Various problems arise, such as determining the minimum number of brushes that are required to clean the entire graph. This number is called the brushing number. Here, we study a new variant of the brushing problem in which one vertex is cleaned at a time, but more than one brush may traverse a dirty edge. In particular, we obtain results on the brushing number of Cartesian products of graphs and trees, as well as upper and lower bounds on the brushing number in the general case.