Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-11T01:51:35.641Z Has data issue: false hasContentIssue false
  • English
  • Français

Lazy Cops and Robbers on Hypercubes

Part of: Graph theory

Published online by Cambridge University Press:  29 January 2015

DEEPAK BAL ,
ANTHONY BONATO ,
WILLIAM B. KINNERSLEY  and
PAWEŁ PRAŁAT
DEEPAK BAL
Affiliation:
Department of Mathematics, Ryerson University, Toronto, ON, Canada, M5B 2K3 (e-mail: deepak.c.bal@ryerson.ca, abonato@ryerson.ca, pralat@ryerson.ca)
ANTHONY BONATO
Affiliation:
Department of Mathematics, Ryerson University, Toronto, ON, Canada, M5B 2K3 (e-mail: deepak.c.bal@ryerson.ca, abonato@ryerson.ca, pralat@ryerson.ca)
WILLIAM B. KINNERSLEY
Affiliation:
Department of Mathematics, University of Rhode Island, Kingston, RI, USA, 02881 (e-mail: billk@mail.uri.edu)
PAWEŁ PRAŁAT
Affiliation:
Department of Mathematics, Ryerson University, Toronto, ON, Canada, M5B 2K3 (e-mail: deepak.c.bal@ryerson.ca, abonato@ryerson.ca, pralat@ryerson.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By coupling the probabilistic method with a potential function argument, we improve on the existing lower bounds for the lazy cop number of hypercubes.

MSC classification

Primary: 05C57: Games on graphs
Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 6 , November 2015 , pp. 829 - 837
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aigner, M. and Fromme, M. (1984) A game of cops and robbers. Discrete Appl. Math. 8 1–12.CrossRefGoogle Scholar
[2] Alon, N. and Mehrabian, A. Chasing a fast robber on planar graphs and random graphs. J. Graph Theory, to appear.Google Scholar
[3] Baird, W. and Bonato, A. (2012) Meyniel's conjecture on the cop number: A survey. J. Combinatorics 3 225–238.CrossRefGoogle Scholar
[4] Bonato, A. (2011) Catch me if you can: Cops and Robbers on graphs. In Proc. 6th International Conference on Mathematical and Computational Models: ICMCM'11.Google Scholar
[5] Bonato, A. (2012) WHAT IS . . . Cop Number? Notices Amer. Math. Soc. 59 1100–1101.CrossRefGoogle Scholar
[6] Bonato, A. and Chiniforooshan, E. (2009) Pursuit and evasion from a distance: algorithms and bounds. In Proc. ANALCO'09.CrossRefGoogle Scholar
[7] Bonato, A., Chiniforooshan, E. and Prałat, P. (2010) Cops and Robbers from a distance. Theoret. Comput. Sci. 411 3834–3844.CrossRefGoogle Scholar
[8] Bonato, A., Finbow, S., Gordinowicz, P., Haidar, A., Kinnersley, W. B., Mitsche, D., Prałat, P. and Stacho, L. (2013) The robber strikes back. In Proc. International Conference on Computational Intelligence, Cyber Security and Computational Models: ICC3.CrossRefGoogle Scholar
[9] Bonato, A. and Nowakowski, R. J. (2011) The Game of Cops and Robbers on Graphs, AMS.CrossRefGoogle Scholar
[10] Dudek, A., Gordinowicz, P. and Prałat, P. (2014) Cops and Robbers playing on edges. J. Combinatorics 5 131–153.CrossRefGoogle Scholar
[11] Frieze, A., Krivelevich, M. and Loh, P. (2012) Variations on Cops and Robbers. J. Graph Theory 69 383–402.CrossRefGoogle Scholar
[12] Kehagias, A., Mitsche, D. and Prałat, P. (2013) Cops and Invisible Robbers: The cost of drunkenness. Theoret. Comput. Sci. 481 100–120.CrossRefGoogle Scholar
[13] Kehagias, A. and Prałat, P. (2012) Some Remarks on Cops and Drunk Robbers. Theoret. Comput. Sci. 463 133–147.CrossRefGoogle Scholar
[14] Nowakowski, R. J. and Winkler, P. (1983) Vertex-to-vertex pursuit in a graph. Discrete Math. 43 235–239.CrossRefGoogle Scholar
[15] Offner, D. and Okajian, K. (2014) Variations of Cops and Robber on the hypercube. Australasian J. Combinatorics 59 229–250.Google Scholar
[16] Quilliot, A. (1978) Jeux et pointes fixes sur les graphes. Thèse de 3ème cycle, Université de Paris VI, pp. 131–145.Google Scholar
[17] West, D. B. (2001) Introduction to Graph Theory, second edition, Prentice Hall.Google Scholar