Manipulative Waiters with Probabilistic Intuition

For positive integers n and q and a monotone graph property A, we consider the two player, perfect information game WC(n, q,A), which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client, q + 1 edges of the complete graph Kn which have not been offered previously. Client then chooses one of these edges which he keeps and the remaining q edges go back to Waiter. If at the end of the game, the graph which consists of the edges chosen by Client satisfies the property A, then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker-Chooser games) for a variety of natural graph theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameter q is close to n and is reminiscent of phase transition phenomena in random graphs. Namely, we prove that if q ≤ (1− ε)n, then Client can avoid connected components of order cε−2 lnn for some absolute constant c > 0, whereas, for q ≥ (1 + ε)n, Waiter can force a giant, linearly sized, connected component in Client’s graph. We also prove that Waiter can force Client’s graph to be pancyclic for every q ≤ cn, where c > 0 is an appropriate constant.