Shifted de Bruijn Graphs

Abstract

We are studying a generalization of the de Bruijn graphs, with applications to storage. We use spectral methods to enumerate the Euler circuits in this graph, which correspond to (very long) strings accessing every string of fixed length exactly once, with the reader reset at regular intervals. We prove that, when the alphabet is of size q, the subwords considered are of length n and a new reader is initiated every k letters, there are exactly $(q^{k})!^{q^{n} }/q^{k+n}$ such exhaustive words. The enumeration generalizes classic results by Tutte, and relates crucially to subtree enumeration in large networks.