Memoryless Rules for Achlioptas Processes

In an Achlioptas process two random pairs of {1, . . . , n} arrive in each round and the player has to choose one of them. We study the very restrictive version where player’s decisions cannot depend on the previous history and only one vertex from the two random edges is revealed. We prove that the player can create a giant component in (2 √ 5 − 4 + o(1))n = (0.4721 . . . + o(1))n rounds and this is best possible. On the other hand, if the player wants to delay the appearence of a giant, then the optimal bound is (1/2 + o(1))n, the same as in the Erdős-Rényi model.