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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N^-(v_i)=\{j: v_j v_i \in E {\rm and} j<\,i\}}$$\end{document} ; finally let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b_{\alpha}(G)=\sum_{i=1}^n {\rm max}\{|N^+(v_i)|-|N^-(v_i)|,0\}}$$\end{document}. The Broom number is given by B(G) =  maxαbα(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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n(d+2\sqrt{d \ln 2})/4}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{n}{4}(d+\Theta(\sqrt{d}))}$$\end{document}.