 
Summary: Cleaning dregular graphs with brushes
Noga Alon
Pawel Pralat
Nicholas Wormald
Abstract
A model for cleaning a graph with brushes was recently introduced. We consider the
minimum number of brushes needed to clean dregular graphs in this model, focusing on
the asymptotic number for random dregular graphs. We use a degreegreedy algorithm
to clean a random dregular graph on n vertices (with dn even) and analyze it using the
differential equations method to find the (asymptotic) number of brushes needed to clean
a random dregular graph using this algorithm (for fixed d). We further show that for any
dregular graph on n vertices at most n(d + 1)/4 brushes suffice, and prove that for fixed
large d, the minimum number of brushes needed to clean a random dregular graph on n
vertices is asymptotically almost surely n
4 (d + o(d)), thus solving a problem raised in [15].
1 Introduction
The cleaning model, introduced in [13, 14], is a combination of the chipfiring game and edge
searching on a simple finite graph. Initially, every edge and vertex of a graph is dirty and a
fixed number of brushes start on a set of vertices. At each step, a vertex v and all its incident
edges which are dirty may be cleaned if there are at least as many brushes on v as there are
