On Embedding 2-Dimensional Toroidal Grids into de Bruijn Graphs with Clocked Congestion One

For integers m; d; D with m 3; d 2; and D 2, let T(m) be a 2{dimensional quadratic toroidal grid with side length m and let B(d;D) be the base d, dimension D de Bruijn graph; assume that jT(m)j = jB(d;D)j. The starting point for our investigations is the observation that, for m; D even, embeddings f : T(m) ! B(d; D) with load 1, expansion 1, and dilation D=2 can easily be found (and have previously been described in the literature). In the present paper, we pose the question whether or not there exist embeddings f : T(m) ! B(d; D) with these properties and with clocked congestion 1. We prove results implying a positive answer to this question when d is greater than two. For d = 2, we do not have a complete answer, but present partial results.