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

We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these parameters is sublinear in the number of vertices of G 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>0 every n-vertex graph with minimum degree (3/4 + γ)n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large.