Efficient embeddings of ternary trees into hypercubes

In this paper we present efficient graph embeddings for complete k-ary trees into boolean hypercubes. In particular, we describe an efficient embedding of a complete ternary tree (k = 3) of height h into a hypercube, which achieves dilation 3 and expansion Θ(1.0104...h). The previously best-known result is dilation 2 and expansion Θ(1.333...h). Our embedding achieves exponentially better expansion at the cost of an increase of 1 in the dilation. We also describe efficient embeddings of k-ary trees into hypercubes when k = 2p * 3q for some p, q > 0 such that the embeddings achieve small constant dilation.