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

Abstract

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 b<n/2−1. In this paper, we show that OBreaker has a winning strategy whenever b>5n/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.