Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs

We establish relations between the bandwidth and the treewidth of bounded degree graphs GG, and relate these parameters to the size of a separator of GG as well as the size of an expanding subgraph of GG. Our results imply that if one of these parameters is sublinear in the number of vertices of GG then so are all the others. This implies for example that graphs of fixed genus have sublinear bandwidth or, more generally, a corresponding result for graphs with any fixed forbidden minor. As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each γ>0γ>0 every nn-vertex graph with minimum degree (34+γ)n contains a copy of every bounded-degree planar graph on nn vertices if nn is sufficiently large.