Cleaning Random d-Regular Graphs with Brooms

Abstract

A model for cleaning a graph with brushes was recently introduced. Let α = (v ₁, v ₂, . . . , v _n ) be a permutation of the vertices of G; for each vertex v _i let ${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}}$ and ${N^-(v_i)=\{j: v_j v_i \in E {\rm and} j<\,i\}}$ ; finally let ${b_{\alpha}(G)=\sum_{i=1}^n {\rm max}\{|N^+(v_i)|-|N^-(v_i)|,0\}}$. The Broom number is given by B(G) =  max _α b _α (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 d-regular 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 ${n(d+2\sqrt{d \ln 2})/4}$ 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 ${\frac{n}{4}(d+\Theta(\sqrt{d}))}$.