Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**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)

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 G_{0},G_{1},... 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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