Cleaning Regular Graphs with Brushes

A model for cleaning a graph with brushes was recently introduced. We consider the minimum number of brushes needed to clean $d$-regular graphs in this model, focusing on the asymptotic number for random $d$-regular graphs. We use a degree-greedy algorithm to clean a random $d$-regular 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 $d$-regular graph using this algorithm (for fixed $d$). We further show that for any $d$-regular 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 $d$-regular graph on $n$ vertices is asymptotically almost surely $\frac{n}{4}(d+o(d))$, thus solving a problem raised in [M.E. Messinger, R.J. Nowakowski, P. Pralat, and N. Wormald, Cleaning random d-regular graphs with brushes using a degree-greedy algorithm, in Combinatorial and Algorithmic Aspects of Networking, Lecture Notes in Comput. Sci. 4852, Springer, Berlin-Heidelberg, 2007, pp. 13-26].