Waiter-Client and Client-Waiter Hamiltonicity games on random graphs

We study two types of two player, perfect information games with no chance moves, played on the edge set of the binomial random graph ${\mathcal G}(n,p)$. In each round of the $(1 : q)$ Waiter-Client Hamiltonicity game, the first player, called Waiter, offers the second player, called Client, $q+1$ edges of ${\mathcal G}(n,p)$ which have not been offered previously. Client then chooses one of these edges, which he claims, and the remaining $q$ edges go back to Waiter. Waiter wins this game if by the time every edge of ${\mathcal G}(n,p)$ has been claimed by some player, the graph consisting of Client's edges is Hamiltonian; otherwise Client is the winner. Client-Waiter games are defined analogously, the main difference being that Client wins the game if his graph is Hamiltonian and Waiter wins otherwise. In this paper we determine a sharp threshold for both games. Namely, for every fixed positive integer $q$, we prove that the smallest edge probability $p$ for which a.a.s. Waiter has a winning strategy for the $(1 : q)$ Waiter-Client Hamiltonicity game is $(1 + o(1)) \log n/n$, and the smallest $p$ for which a.a.s. Client has a winning strategy for the $(1 : q)$ Client-Waiter Hamiltonicity game is $(q + 1 + o(1)) \log n/n$.