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

Date and Time: Thursday, January 20, 2011, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Reto Spöhel (MPI Saarbrücken)

Creating small subgraphs in the Achlioptas process.

The standard paradigm for online power of two choices problems in random graphs is the Achlioptas process. Here we consider the following natural generalization: Starting with $G_0$ as the empty graph on $n$ vertices, in every step a set of $r$ edges is drawn uniformly at random from all edges that have not been drawn in previous steps. From these, one edge has to be selected, and the remaining $r-1$ edges are discarded. Thus after $N$ steps, we have seen $rN$ edges, and selected exactly $N$ out of these to create a graph $G_N$.

The problem of avoiding copies of some given fixed graph $F$ for as long as possible in this process (with $r$= const.) was first studied by Krivelevich, Loh, and Sudakov (2009), and solved completely by Torsten Mütze, Henning Thomas, and myself.

In this talk we consider the opposite problem of creating a copy of $F$ as quickly as possible. As we shall see, this problem behaves quite differently - in particular, it is only interesting if $r$ is assumed to be a growing function of $n$. We give general bounds on the threshold of this problem, and derive exact thresholds for trees and cycles of arbitrary size, and for $F=K_4$.

Joint work with Michael Krivelevich (Tel Aviv University).

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