On embedding 2-dimensional toroidal grids into de Bruijn graphs with clocked congestion one

Abstract

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 |T(m)|=|B(d, D)|. 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.