## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, March 07, 2006, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Konstantinos Panagiotou

## Extremal Subgraphs of Random Graphs

For a graph G, let ET(G) denote the maximum number of edges in a triangle-free subgraph (not necessarily induced) of G, and let EB(G) be the maximum number of edges in a bipartite subgraph of G.

Of course, we always have ET(G) ≥ EB(G), but the general intuition -- guided by various known results -- suggests that, for dense enough graphs, these two parameters will typically be equal.

In 1990, Babai, Simonovits and Spencer studied these parameters for random graphs G(n,p) and confirmed this intuition for dense graphs. In particular, they proved that there is a (small) positive constant c such that, for p ≥ 1/2 - c, with high probability we have ET(G(n,p)) = EB(G(n,p)).

Babai, Simonovits and Spencer asked whether this result could be extended to cover all constant values of p. In this talk we answer this question affirmatively and show that the above property in fact holds whenever p=p(n) ≥ n-c, for some fixed c > 0.

This is joint work with Graham Brightwell and Angelika Steger.

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