Cartesian Products of Graphs as Spanning Subgraphs of De Bruijn Graphs (Extended Abstract)

For Cartesian products G = G1. .. Gm (m 2) of nontrivial connected graphs Gi and the n-dimensional base B de Bruijn graph D = DB(n), we investigate whether or not there exists a spanning subgraph of D which is isomorphic to G. We show that G is never a spanning subgraph of D when n is greater than three or when n equals three and m is greater than two. For n = 3 and m = 2, we can show for wide classes of graphs that G cannot be a spanning subgraph of D. In particular, these non-existence results imply that DB(n) never contains a torus (i.e., the Cartesian product of m 2 cycles) as a spanning subgraph when n is greater than two. For n = 2 the situation is quite diierent: we present a suucient condition for a Cartesian product G to be a spanning subgraph of D = DB(2). As one of the corollaries we obtain that a torus G = G1. .. Gm is a spanning subgraph of D = DB(2) provided that jGj = jDj and that the Gi are even cycles of length 4. In addition we apply our results to obtain embeddings of relatively small dilation of popular processor networks (as tori, meshes and hypercubes) into de Bruijn graphs of xed small base.