TY - JOUR

T1 - Playing to retain the advantage

AU - Alon, Noga

AU - Hefetz, Dan

AU - Krivelevich, Michael

PY - 2010/7

Y1 - 2010/7

N2 - 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].

AB - 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].

UR - http://www.scopus.com/inward/record.url?scp=77955763176&partnerID=8YFLogxK

U2 - 10.1017/S0963548310000064

DO - 10.1017/S0963548310000064

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AN - SCOPUS:77955763176

SN - 0963-5483

VL - 19

SP - 481

EP - 491

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 4

ER -