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 K_n 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 gamȩ 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 components of order cϵ^-2 ln n for some absolute constant c > 0, whereas for q ≤(1 - ϵ)n, Waiter can force a giant, linearly sized component in Client's graph. In the second part of the paper, we prove that Waiter can force Client's graph to be pancyclic for every q ≤ cn, where c > 0 is an appropriate constant. Note that this behaviour is in stark contrast to the threshold for pancyclicity and Hamiltonicity of random graphs.