On two problems regarding the Harniltonian cycle game

We consider the fair Harniltonian cycle Maker-Breaker game, played on the edge set of the complete graph K_n on n vertices. It is known that Maker wins this game if n is sufficiently large. We are interested in the minimum number of moves needed for Maker in order to win the Hamiltonian cycle game, and in the smallest n for which Maker has a winning strategy for this game. We prove the following results: (1) If n is sufficiently large, then Maker can win the Hamiltonian cycle game within n + 1 moves. This bound is best possible and it settles a question of Hefetz, Krivelevich, Stojaković and Szabó; (2) If n ≥ 29, then Maker can win the Hamiltonian cycle game. This improves the previously best bound of 600 due to Papaioannou.