AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 43 (2009), Pages 237–251 Cleaning random graphs with brushes

A model for cleaning a graph with brushes was recently introduced. In this paper, we consider the minimum number of brushes needed to clean a random graph G(n, p = d/n) in this model, the so-called brush number. We show that the brush number of a random graph on n vertices is asymptotically almost surely (a.a.s.) dn 4 (1 + o(1)) if the average degree is tending to infinity with n. For a constant d > 1, various upper and lower bounds are studied. For d ≤ 1, we show that the number of brushes needed is a.a.s. n 4 (1 − exp(−2d))(1 + o(1)) and compute the probability that it attains its natural lower bound.