Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, April 24, 2012, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Daniel Johannsen (Tel Aviv University)

Embedding spanning trees in random graphs

A classical result by Erdős and Rényi states that asymptotically almost surely (a.a.s.) a random graph G in the G(n,p) model is connected if the expected degree pn of its vertices is slightly larger than log(n). Naturally, this also implies that such a graph contains a spanning tree.

Recently, Krivelevich showed that if pn is at least of order n^ε with ε>0 then G a.a.s. contains any single fixed spanning tree with bounded maximum degree.

We study the question for which values of pn the random graph G contains every spanning tree with bounded maximum degree.

To this end, we investigate sparse graphs on n vertices which satisfy certain natural expansion properties. We show that each graph with these properties does indeed contain every n-vertex tree with bounded maximum degree. We then see that a random graph drawn according to the G(n,p) model with pn at least of order n^2/3log²n a.a.s. has the desired expansion properties and therefore contains every n-vertex tree with bounded maximum degree.

Joint work with Michael Krivelevich and Wojciech Samotij.

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