Global Maker-Breaker games on sparse graphs

Dan Hefetz, Michael Krivelevich, Miloš Stojaković, Tibor Szabó

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we consider Maker-Breaker games, played on the edges of sparse graphs. For a given graph property P we seek a graph (board of the game) with the smallest number of edges on which Maker can build a subgraph that satisfies P. In this paper we focus on global properties. We prove the following results: (1) for the positive minimum degree game, there is a winning board with n vertices and about 10n/7 edges, on the other hand, at least 11n/8 edges are required; (2) for the spanning k-connectivity game, there is a winning board with n vertices and (1+ok(1))kn edges; (3) for the Hamiltonicity game, there is a winning board of constant average degree; (4) for a tree T on n vertices of bounded maximum degree δ, there is a graph G on n vertices and at most f(δ).n edges, on which Maker can construct a copy of T. We also discuss biased versions of these games and argue that the picture changes quite drastically there.

Original languageEnglish
Pages (from-to)162-177
Number of pages16
JournalEuropean Journal of Combinatorics
Volume32
Issue number2
DOIs
StatePublished - Feb 2011

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