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 0(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. © 1999 John Wiley & Sons, Inc. Networks 33: 71-77, 1999