Embeddings of shuffle-like graphs in hypercubes

Abstract

Translate Article

Let G and H be two simple undirected graphs. An embedding of the graph G in the graph H is an injective mapping f from the vertices of G into the vertices of H together with a mapping Pf of edges of G into paths in H. The dilation of the embedding is the maximum taken over all the lengths of the paths Pf(x,y) associated with the edges (x,y) of G.One challenge pointed out in [9] is to find embeddings of the de Bruijn graph in the hypercube of the same order which have a low dilation. For a de Bruijn graph of diameter D we give an embedding in a hypercube of the same diameter of dilation 2[D/51], and determine the edge-congestion and vertex-congestion of this embedding. Similar results are given for the shuffle-exchange graphs.