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
On balanced coloring games in random graphs
European Journal of Combinatorics
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For a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is a natural generalization of what is known in the literature as an Achlioptas process (the original version has r = 2), which has been studied by many researchers, mainly in the context of delaying or accelerating the appearance of the giant component. In this article, we investigate the small subgraph problem for Achlioptas processes. That is, given a fixed graph H, we study whether there is an online algorithm that substantially delays or accelerates a typical appearance of H, compared to its threshold of appearance in the random graph G(n, M). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than the constant factor r, so we concentrate on the task of avoiding H. We determine thresholds for the avoidance of all cycles Ct, cliques Kt, and complete bipartite graphs Kt,t, in every Achlioptas process with parameter r ge 2. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009