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

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, November 03, 2020, 12:15 pm

**Duration**: 30 minutes

**Location**: Zoom: conference room

**Speaker**: Nemanja Draganic

## Rolling backwards can move you forward: on embedding problems in sparse expanders

We develop a general embedding method based on the Friedman-Pippenger tree embedding technique (1987) and its algorithmic version, enhanced with a roll-back idea allowing to sequentially retrace previously performed embedding steps. This proves to be a powerful tool for embedding graphs of large girth into expander graphs. As an application of this method, we settle two problems:

-For a graph $H$, we denote by $H^q$ the graph obtained from $H$ by subdividing its edges with $q-1$ vertices each. We show that the $k$-size-Ramsey number $\hat{R}_k(H^q)$ satisfies $\hat{R}_k(H^q)=O(qn)$ for every bounded degree graph $H$ on $n$ vertices and for $q=\Omega(\log n)$, which is optimal up to a constant factor. This settles a conjecture of Pak (2002).

-We give a deterministic, polynomial time algorithm for finding vertex-disjoint paths between a given number of pairs of vertices (which is optimal up to a constant factor) in a strong expander graph. The result answers the offline version of a question of Alon and Capalbo (2007).

Joint work with Michael Krivelevich And Rajko Nenadov.

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