Playing to Retain the Advantage

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, Luczak and Rödl [6].