One-to-Many Embeddings of Hypercubes into Cayley Graphs Generated by Reversals

Abstract.

Among Cayley graphs on the symmetric group, some that have a noticeably low diameter relative to the degree of regularity are examples such as the ``star'' network and the ``pancake'' network, the latter a representative of a variety of Cayley graphs generated by reversals. These diameter and degree conditions make these graphs potential candidates for parallel computation networks. Thus it is natural to investigate how well they can simulate other standard parallel networks, in particular hypercubes. For this purpose, constructions have previously been given for low dilation embeddings of hypercubes into star networks. Developing this theme further, in this paper we construct especially low dilation maps (e.g., with dilation 1, 2, 3, or 4) of hypercubes into pancake networks and related Cayley graphs generated by reversals. Whereas obtaining such results by the use of ``traditional'' graph embeddings (i.e., one-to-one or many-to-one embeddings) is sometimes difficult or impossible, we achieve many of these results by using a nontraditional simulation technique known as a ``one-to-many'' graph embedding. That is, in such embeddings we allow each vertex in the guest (i.e., domain) graph to be associated with some nonempty subset of the vertex set of the host (i.e., range) graph, these subsets satisfying certain distance and connection requirements which make the simulation possible.