## 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: Thursday, March 20, 2014, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Ueli Peter

## Universality of random graphs

A graph $G$ is unversal with respect to a family of graphs $\mathcal{H}$ if every $H\in \mathcal{H}$ can be embedded into $G$. In this talk we present a partitioning lemma that helps to embed fixed graphs into the random graph $G(n;p)$. As applications we show how to derive new bounds for the edge probability $p$ for which a typical random graph $G(n; p)$ is universal with respect to some families of graphs. In particular, we improve the result of Johannsen, Krivelevich and Samotij for the universality of $G(n; p)$ with respect to the family of spanning bounded degree forests. In addition, we improve a result of Dellamonica, Kohayakawa, Rodl and Rucinski for the universality of $G(n; p)$ with respect to the family of bounded degree graphs with density $d$ much smaller than the maximum degree. Joint work with Asaf Ferber and Rajko Nenadov

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