TY - JOUR

T1 - Cleaning regular graphs with brushes

AU - Alon, Noga

AU - Prałat, Paweł

AU - Wormald, Nicholas

PY - 2008

Y1 - 2008

N2 - 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 [M.E. Messinger, R.J. Nowakowski, P. Prałat, 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].

AB - 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 [M.E. Messinger, R.J. Nowakowski, P. Prałat, 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].

KW - Cleaning process

KW - Degree-greedy algorithm

KW - Differential equations method

KW - Random d-regular graphs

UR - http://www.scopus.com/inward/record.url?scp=62549147212&partnerID=8YFLogxK

U2 - 10.1137/070703053

DO - 10.1137/070703053

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AN - SCOPUS:62549147212

SN - 0895-4801

VL - 23

SP - 233

EP - 250

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -