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Playing to Retain the Advantage

Published online by Cambridge University Press:  19 March 2010

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)

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
Copyright
Copyright © Cambridge University Press 2010

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References

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