On Hypercubes in de Bruijn Graphs

We prove that the hypercube of odd dimension 2k + 1 is a subgraph of the de Bruijn graph of alphabet size d and diameter 2 if and only if d ≥ 3 · 2k-1. This complements previous results of Heydemann, Opatrny, and Sotteau (1994) and Andreae et al. (1995), thus yielding a complete solution of the problem of determining, for all integers m, n ≥ 2, the least number d = d(m, n) for which the hypercube of dimension m is a subgraph of the de Bruijn graph of alphabet size d and diameter n.