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

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, November 25, 2008, 12:15 pm

**Duration**: This information is not available in the database

**Location**: OAT S15/S16/S17

**Speaker**: Torsten Mütze

## Small subgraphs in random graphs and the power of multiple choices: The online case

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 $G_N$.

In a 2007 paper by Krivelevich, Loh, and Sudakov, the problem of avoiding a copy of some fixed graph $F$
in $G_N$ for as long as possible is considered, and a threshold result is derived for some special cases.
Moreover, the authors conjecture a general threshold formula for arbitrary graphs $F$. We disprove this
conjecture and give the complete solution of the problem by deriving explicit threshold functions
$N_0(F,r,n)$ for arbitrary graphs $F$ and any fixed integer $r$.

Joint work with Reto Spöhel and Henning Thomas.

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