Abstract
In the recently introduced model for cleaning a graph with brushes, we use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even). We then use a differential equations method to find the (asymptotic) number of brushes needed to clean a random d-regular graph using this algorithm. As well as the case for general d, interesting results for specific values of d are examined. We also state various open problems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bollobás, B.: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European Journal of Combinatorics 1, 311–316 (1980)
Bollobás, B.: Random graphs, Combinatorics. In: Temperley, H.N.V. (ed.) London Mathematical Society Lecture Note Series, vol. 52, pp. 80–102. Cambridge University Press, Cambridge (1981)
Kim, J.H., Wormald, N.C.: Random matchings which induce Hamilton cycles and hamiltonian decompositions of random regular graphs. Journal of Combinatorial Theory, Series B 81, 20–44 (2001)
Łuczak, T.: Sparse random graphs with a given degree sequence. In: Frieze, A., Łuczak, T. (eds.) Random Graphs, vol. 2, pp. 165–182. Wiley, New York (1992)
McKeil, S.: Chip Firing Cleaning Processes. MSc Thesis, Dalhousie University (2007)
Messinger, M.E., Nowakowski, R.J., Prałat, P.: Cleaning a Network with Brushes. Theoretical Computer Science (accepted)
Monagan, M.B., Geddes, K.O., Heal, K.M., Labahn, G., Vorkoetter, S.M., McCarron, J., DeMarco, P.: Maple 10 Programming Guide. Maplesoft, Waterloo, Canada (2005)
Robinson, R.W., Wormald, N.C.: Almost all cubic graphs are hamiltonian. Random Structures and Algorithms 3, 117–125 (1992)
Wormald, N.C.: Analysis of greedy algorithms on graphs with bounded degrees. EuroComb 2001. Discrete Mathematics 273, 235–260 (2003)
Wormald, N.C.: Models of random regular graphs. In: Lamb, J.D., Preece, D.A. (eds.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 276, pp. 239–298. Cambridge University Press, Cambridge (1999)
Wormald, N.C.: The asymptotic connectivity of labelled regular graphs. Journal of Combinatorial Theory, Series B 31, 156–167 (1981)
Wormald, N.C.: The differential equation method for random graph processes and greedy algorithms. In: Karoński, M., Prömel, H.J. (eds.) Lectures on Approximation and Randomized Algorithms. PWN, Warsaw, pp. 73–155 (1999)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Messinger, ME., Prałat, P., Nowakowski, R.J., Wormald, N. (2007). Cleaning Random d-Regular Graphs with Brushes Using a Degree-Greedy Algorithm. In: Janssen, J., Prałat, P. (eds) Combinatorial and Algorithmic Aspects of Networking. CAAN 2007. Lecture Notes in Computer Science, vol 4852. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77294-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-77294-1_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77293-4
Online ISBN: 978-3-540-77294-1
eBook Packages: Computer ScienceComputer Science (R0)