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Playing to Retain the Advantage

Published online by Cambridge University Press:  19 March 2010

NOGA ALON ,
DAN HEFETZ  and
MICHAEL KRIVELEVICH
NOGA ALON
Affiliation:
Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: nogaa@tau.ac.il)
DAN HEFETZ
Affiliation:
Institute of Theoretical Computer Science, ETH Zurich, CH-8092 Switzerland (e-mail: dan.hefetz@inf.ethz.ch)
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel (e-mail: krivelev@post.tau.ac.il)
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Abstract

Let P be a monotone increasing graph property, let G = (V, E) be a graph, and let q be a positive integer. In this paper, we study the (1: q) Maker–Breaker game, played on the edges of G, in which Maker's goal is to build a graph that satisfies the property P. It is clear that in order for Maker to have a chance of winning, G itself must satisfy P. We prove that if G satisfies P in some strong sense, that is, if one has to delete sufficiently many edges from G in order to obtain a graph that does not satisfy P, then Maker has a winning strategy for this game. We also consider a different notion of satisfying some property in a strong sense, which is motivated by a problem of Duffus, Łuczak and Rödl [6].

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 4 , July 2010 , pp. 481 - 491
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Alon, N. (1995) A note on network reliability. In Discrete Probability and Algorithms (Aldous, D., Diaconis, P., Spencer, J. and Steele, J. M., eds), Vol. 72 of IMA Volumes in Mathematics and its Applications, Springer, pp. 11–14.CrossRefGoogle Scholar
[2]Alon, N. and Shapira, A. (2005) Every monotone graph property is testable. In Proc. 37 ACM STOC, Baltimore, ACM Press, pp. 128–137. Also: SICOMP (Special Issue of STOC'05) 38 (2008) 505–522.Google Scholar
[3]Beck, J. (1982) Remarks on positional games. Acta Math. Acad. Sci. Hungar. 40 65–71.Google Scholar
[4]Bednarska, M. and Łuczak, T. (2000) Biased positional games for which random strategies are nearly optimal. Combinatorica 20 477–488.Google Scholar
[5]Chvátal, V. and Erdős, P. (1978) Biased positional games. Ann. Discrete Math. 2 221–228.Google Scholar
[6]Duffus, D., Łuczak, T. and Rödl, V. (1998) Biased positional games on hypergraphs. Studia Sci. Math. Hungar. 34 141–149.Google Scholar
[7]Hefetz, D., Krivelevich, M., Stojaković, M. and Szabó, T. (2008) Planarity, colorability and minor games. SIAM J. Discrete Math. 22 194–212.CrossRefGoogle Scholar
[8]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley, New York.Google Scholar
[9]Johansson, A. (1996) Asymptotic choice number for triangle free graphs. DIMACS Technical Report 91-95. 1196.Google Scholar
[10]Karger, D. (1993) Global min-cuts and other ramifications of a simple min-cut algorithm. In Proc. 4th Annual ACM–SIAM Symposium on Discrete Algorithms (Austin TX), ACM, New York, pp. 21–30.Google Scholar
[11]Lehman, A. (1964) A solution of the Shannon switching game. J. Soc. Indust. Appl. Math. 12 687–725.CrossRefGoogle Scholar
[12]Nash-Williams, C. S. J. A. (1961) Edge disjoint spanning trees of finite graphs. J. London Math. Soc. 36 445–450.CrossRefGoogle Scholar
[13]Tutte, W. T. (1961) On the problem of decomposing a graph into n connected factors. J. London Math. Soc. 36 221–230.CrossRefGoogle Scholar
[14]West, D. B. (2001) Introduction to Graph Theory, Prentice-Hall.Google Scholar