On Two Problems Regarding the Hamiltonian Cycle Game

We consider the fair Hamiltonian 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, Stojakovic and Szabo; (2) If $n \geq 29$, then Maker can win the Hamiltonian cycle game. This improves the previously best bound of $600$ due to Papaioannou.