TY - JOUR

T1 - Playing to retain the advantage

AU - Hefetz, Dan

AU - Alon, Noga

AU - Krivelevich, Michael

PY - 2009/8/1

Y1 - 2009/8/1

N2 - Let P be a monotone decreasing 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 does not satisfy the property P. It is clear that in order for Maker to have a chance of winning, G must not satisfy P. We prove that if G is far from satisfying P, that is, if one has to delete sufficiently many edges from G in order to obtain a graph that satisfies P, then Maker has a winning strategy for this game. We also consider a different notion of being far from satisfying some property, which is motivated by a problem of Duffus, Łuczak and Rödl [D. Duffus, T. Łuczak and V. Rödl, Biased positional games on hypergraphs, Studia Scientarum Matematicarum Hung. 34 (1998), 141-149].

AB - Let P be a monotone decreasing 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 does not satisfy the property P. It is clear that in order for Maker to have a chance of winning, G must not satisfy P. We prove that if G is far from satisfying P, that is, if one has to delete sufficiently many edges from G in order to obtain a graph that satisfies P, then Maker has a winning strategy for this game. We also consider a different notion of being far from satisfying some property, which is motivated by a problem of Duffus, Łuczak and Rödl [D. Duffus, T. Łuczak and V. Rödl, Biased positional games on hypergraphs, Studia Scientarum Matematicarum Hung. 34 (1998), 141-149].

KW - Positional games

KW - monotone property

UR - http://www.scopus.com/inward/record.url?scp=67651166769&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2009.07.070

DO - 10.1016/j.endm.2009.07.070

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AN - SCOPUS:67651166769

SN - 1571-0653

VL - 34

SP - 423

EP - 427

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

ER -