Cleaning Random d-Regular Graphs with Brooms

A model for cleaning a graph with brushes was recently introduced. Let α = (v1, v2, . . . , vn) be a permutation of the vertices of G; for each vertex vi let N (vi) = {j : vjvi ∈ E and j > i} and N−(vi) = {j : vjvi ∈ E and j < i}; finally let bα(G) = ∑n i=1 max{|N+(vi)|−|N−(vi)|, 0}. The Broom number is given by B(G) = maxα bα(G). We consider the Broom number of d-regular graphs, focusing on the asymptotic number for random d-regular graphs. Various lower and upper bounds are proposed. To get an asymptotically almost sure lower bound 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 (for fixed d). We further show that for any dregular graph on n vertices there is a cleaning sequence such at least n(d + 1)/4 brushes are needed to clean a graph using this sequence. For an asymptotically almost sure upper bound, the pairing model is used to show that at most n(d + 2 √ d ln 2)/4 brushes can be used when a random d-regular graph is cleaned. This implies that for fixed large d, the Broom number of a random d-regular graph on n vertices is asymptotically almost surely n4 (d+Θ( √ d)).