Fast winning strategies in Maker-Breaker games

Dan Hefetz, Michael Krivelevich, Miloš Stojaković, Tibor Szabó

Research output: Contribution to journalArticlepeer-review

45 Scopus citations


We consider unbiased Maker-Breaker games played on the edge set of the complete graph Kn on n vertices. Quite a few such games were researched in the literature and are known to be Maker's win. Here we are interested in estimating the minimum number of moves needed for Maker in order to win these games. We prove the following results, for sufficiently large n: (1)Maker can construct a Hamilton cycle within at most n + 2 moves. This improves the classical bound of 2n due to Chvátal and Erdo{combining double acute accent}s [V. Chvátal, P. Erdo{combining double acute accent}s, Biased positional games, Ann. Discrete Math. 2 (1978) 221-228] and is almost tight;(2)Maker can construct a perfect matching (for even n) within n / 2 + 1 moves, and this is tight;(3)For a fixed k ≥ 3, Maker can construct a spanning k-connected graph within (1 + o (1)) k n / 2 moves, and this is obviously asymptotically tight. Several other related results are derived as well.

Original languageEnglish
Pages (from-to)39-47
Number of pages9
JournalJournal of Combinatorial Theory. Series B
Issue number1
StatePublished - Jan 2009
Externally publishedYes


  • Hamilton cycle
  • Maker-Breaker games
  • Perfect matching
  • k-connected graph


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