The treewidth and pathwidth of hypercubes

The d  -dimensional hypercube, HdHd, is the graph on 2d2d vertices, which correspond to the 2d2dd  -vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate. The notion of Hamming graphs (denoted by Kqd) generalizes the notion of hypercubes: The vertices correspond to the qdqdd  -vectors where the components are from the set {0,1,2,…,q-1}{0,1,2,…,q-1}, and two of the vertices are adjacent if and only if the corresponding vectors differ in exactly one component. In this paper we show that the pw(Hd)=∑m=0d-1mm2 and the tw(Kqd)=θ(qd/d).