A non-trivial upper bound on the threshold bias of the Oriented-cycle game

Dennis Clemens, Anita Liebenau

Research output: Contribution to journalArticleResearchpeer-review


In the Oriented-cycle game, introduced by Bollobás and Szabó [7], two players, called OMaker and OBreaker, alternately direct edges of Kn. OMaker directs exactly one previously undirected edge, whereas OBreaker is allowed to direct between one and b previously undirected edges. OMaker wins if the final tournament contains a directed cycle, otherwise OBreaker wins. Bollobás and Szabó [7] conjectured that for a bias as large as n-3 OMaker has a winning strategy if OBreaker must take exactly b edges in each round. It was shown recently by Ben-Eliezer, Krivelevich and Sudakov [6], that OMaker has a winning strategy for this game whenever b5n/6+1. Moreover, in case OBreaker is required to direct exactly b edges in each move, we show that OBreaker wins for b≥19n/20, provided that n is large enough. This refutes the conjecture by Bollobás and Szabó.

Original languageEnglish
Pages (from-to)21-54
Number of pages34
JournalJournal of Combinatorial Theory, Series B
Publication statusPublished - Jan 2017


  • Cycles
  • Digraphs
  • Orientation games

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