Embedding bounded bandwidth graphs into ℓ1

We introduce the first embedding of graphs of low bandwidth into ℓ1, with distortion depending only upon the bandwidth. We extend this result to a new graph parameter called tree-bandwidth, which is very similar to (but more restrictive than) treewidth. This represents the first constant distortion embedding of a non-planar class of graphs into ℓ1. Our results make use of a new technique that we call iterative embedding in which we define coordinates for a small number of points at a time.