Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Wednesday, April 20, 2005, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Nick Wormald (Univ. of Waterloo)

Birth control for giants

We consider a problem of Dimitris Achlioptas: initially a graph G has no edges. In each round two edges of K_n are generated independently and uniformly at random. We must select one of those edges and add it to G. Our object is to avoid creating a giant component, for as long as possible. This motivates the following problem. Fix any algorithm that determines which edge we select. Let G_m denote the graph after m rounds. Then G₀,G₁,... forms a random graph process, that evolves from the empty graph to a graph with a giant component and, of course, beyond. For the standard random graphs, the giant component appears after about n/2 edges. For a general class of algorithms we analyse this process and show the existence of a phase transition in the size of the largest component. For the original Achlioptas problem, we also obtain lower bounds on how long we can avoid creating a giant (asymptotically almost surely), of more than 0.8n edges. For the converse problem of trying to create a giant, we have upper bounds of less than 0.35n.

(Joint work with Joel Spencer)

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