Hamiltonicity thresholds in Achlioptas processes

In this article, we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K = K ( n ) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K . For K = o (log n ), the threshold for Hamiltonicity is ${1 + o(1) \over 2K}$ n log n , i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K = ω (log n ) we can essentially waste almost no edges, and create a Hamilton cycle in n + o ( n ) rounds with high probability. Finally, in the intermediate regime where K = Θ (log n ), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010