Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-29T04:41:03.465Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast Strategies In Maker–Breaker Games Played on Random Boards

Published online by Cambridge University Press:  10 September 2012

DENNIS CLEMENS ,
ASAF FERBER ,
MICHAEL KRIVELEVICH  and
ANITA LIEBENAU
DENNIS CLEMENS
Affiliation:
Department of Mathematics and Computer Science, Freie Universität Berlin, Germany (e-mail: d.clemens@fu-berlin.de, liebenau@math.fu-berlin.de)
ASAF FERBER
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: ferberas@post.tau.ac.il, krivelev@post.tau.ac.il)
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: ferberas@post.tau.ac.il, krivelev@post.tau.ac.il)
ANITA LIEBENAU
Affiliation:
Department of Mathematics and Computer Science, Freie Universität Berlin, Germany (e-mail: d.clemens@fu-berlin.de, liebenau@math.fu-berlin.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we analyse classical Maker–Breaker games played on the edge set of a sparse random board G ~ n,p. We consider the Hamiltonicity game, the perfect matching game and the k-connectivity game. We prove that for p(n) ≥ polylog(n)/n the board G ~ n,p is typically such that Maker can win these games asymptotically as fast as possible, i.e., within n+o(n), n/2+o(n) and kn/2+o(n) moves respectively.

Keywords

Primary 05C8005C5791A4391A46
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 6 , November 2012 , pp. 897 - 915
Copyright
Copyright © Cambridge University Press 2012

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]Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, Wiley.Google Scholar
[2]Beck, J. (1982) Remarks on positional games. Acta Math. Acad. Sci. Hungar. 40 65–71.Google Scholar
[3]Beck, J. (2008) Combinatorial Games: Tic-Tac-Toe Theory, Cambridge University Press.CrossRefGoogle Scholar
[4]Ben-Eliezer, I., Krivelevich, M. and Sudakov, B. (2012) The size Ramsey number of a directed path. J.~Combin. Theory Ser. B 102 743–755.Google Scholar
[5]Ben-Shimon, S., Ferber, A., Hefetz, D. and Krivelevich, M. (2011) Hitting time results for Maker–Breaker games. In Proc. 22nd Symposium on Discrete Algorithms: SODA'11, pp. 900–912.Google Scholar
[6]Chvátal, V. and Erdős, P. (1978) Biased positional games. Ann. of Discrete Math. 2 221–228.Google Scholar
[7]Ferber, A. and Hefetz, D. (2011) Winning strong games through fast strategies for weak games. Electron. J. Combin. 18 P144.Google Scholar
[8]Ferber, A. and Hefetz, D. Weak and strong k-connectivity game. Preprint. www.math.tau.ac.il/~ferberas/papersGoogle Scholar
[9]Ferber, A., Hefetz, D. and Krivelevich, M. (2012) Fast embedding of spanning trees in biased Maker–Breaker games. Europ. J. Combin. 33 1086–1099.CrossRefGoogle Scholar
[10]Hefetz, D., Krivelevich, M., Stojaković, M. and Szabó, T. (2009) Fast winning strategies in Maker–Breaker games. J. Combin. Theory Ser. B 99 39–47.Google Scholar
[11]Hefetz, D., Krivelevich, M. and Szabó, T. (2009) Hamilton cycles in highly connected and expanding graphs. Combinatorica 29 547–568.CrossRefGoogle Scholar
[12]Hefetz, D., Krivelevich, M. and Szabó, T. Sharp threshold for the appearance of certain spanning trees in random graphs. Random Struct. Alg., to appear.Google Scholar
[13]Hefetz, D. and Stich, S. (2009) On two problems regarding the Hamilton cycle game. Electron. J. Combin. 16 R28.Google Scholar
[14]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.CrossRefGoogle Scholar
[15]Krivelevich, M. (2010) Embedding spanning trees in random graphs. SIAM J. Discrete Math. 24 1495–1500.Google Scholar
[16]Lehman, A. (1964) A solution to the Shannon switching game. J. Soc. Indust. Appl. Math. 12 687–725.Google Scholar
[17]Stojaković, M. and Szabó, T. (2005) Positional games on random graphs. Random Struct. Alg. 26 204–223.CrossRefGoogle Scholar
[18]West, D. B. (2001) Introduction to Graph Theory, Prentice Hall.Google Scholar