Small congestion embedding of graphs into hypercubes

We consider the problem of embedding graphs into hypercubes with minimal congestion. Kim and Lai showed that for a given N-vertex graph G and a hypercube it is NP-complete to determine whether G is embeddable in the hypercube with unit congestion, but G can be embedded with unit congestion in a hypercube of dimension 6log N if the maximum degree of a vertex in G is no more than 6log N . Bhatt et al. showed that every N-vertex binary tree can be embedded in a hypercube of dimension  log N with O (1 ) congestion. In this paper, we extend the results above and show the following: (1) Every N-vertex graph G can be embedded with unit congestion in a hypercube of dimension 2log N if the maximum degree of a vertex in G is no more than 2log N , and (2) every N-vertex binary tree can be embedded in a hypercube of dimension  log N with congestion at most 5. The former answers a question posed by Kim and Lai. The latter is the first result that shows a simple embedding of a binary tree into an optimal-sized hypercube with an explicit small congestion of 5. This partially answers a question posed by Bhatt et al. The embeddings proposed here are quite simple and can be constructed in polynomial time. q 1999 John Wiley & Sons, Inc. Networks 33: 71–77, 1999