Delaying satisfiability for random 2SAT
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Small subgraphs in random graphs and the power of multiple choices
Journal of Combinatorial Theory Series B
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In this article, we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on nlabeled 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 nand we obtain upper and lower bounds that differ by a multiplicative factor of 3. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010