Embedding spanning graphs into dense and sparse graphs

In this talk we will first survey results which guarantee the existence of

spanning subgraphs in dense graphs. This will lead us to the proof of the

bandwidth-conjecture by Bollobas and Komlos, which states that any graph

with minimum degree at least $(1-1/r+\epsilon)n$ contains every r-chromatic graph

with bounded maximum degree and sublinear bandwidth as a spanning subgraph.

We will then move on to discuss the analogous question for a host graph that

is obtained by starting from a sparse random graph G(n,p) and deleting a

certain portion of the edges incident at every vertex.

This is joint work with J. Boettcher, Y. Kohayakawa and M. Schacht.