Hamiltonicity of the Cross-Join Graph of de Bruijn Sequences

A generalized de Bruijn digraph generalizes a de Bruijn digraph to the case where the number of vertices need not be a pure power of an integer. Hamiltonian cycles in these digraphs thus generalize regular de~Bruijn cycles, and we will thus refer to them simply as de Bruijn cycles. We define the cross-join to be the graph with all de Bruijn cycles as vertices, there is an edge between two of these vertices if one can be obtained from the other via a cross-join operation. We show that the cross-join graph is connected. This in particular means that any regular de Bruijn cycle can be cross-joined repeatedly to reach any other de Bruijn cycle, generalizing a result about regular binary de Bruijn cycles by Mykkeltveit and Szmidt in 2014. Furthermore, we present an algorithm that produces a Hamiltonian path across the cross-join graph, one that we may call a de~Bruijn sequence of de Bruijn sequences.