Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
Cleaning d-regular graphs with brushes Pawel Pralat
 

Summary: Cleaning d-regular 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 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 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 chip-firing 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

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics