Cartesian products of graphs as spanning subgraphs of de Bruijn graphs
For Cartesian products G=G1×⋯× G m (m≥2) of nontrivial connected graphs G i and the n-dimensional base B de Bruijn graph D=D B (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 D B (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 different: we present a sufficient condition for a Cartesian product G to be a spanning subgraph of D=D B (2). As one of the corollaries we obtain that a torus G=G1×⋯× G m is a spanning subgraph of D=D B (2) provided that ∥G∥=∥D∥ and that the G i 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 fixed small base.
Keywordsde Bruijn graphs Cartesian product graph embeddings dilation processor networks parallel image processing and pattern recognition massively parallel computers
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