Hamiltonicity of the Cross-Join Graph of de Bruijn Sequences

TL;DR

This paper studies the structure of the cross-join graph of generalized de Bruijn sequences, proving its connectivity and providing an algorithm to traverse all such sequences via Hamiltonian paths, thus generalizing previous binary results.

Contribution

It establishes the connectivity of the cross-join graph for generalized de Bruijn sequences and introduces an algorithm for Hamiltonian paths through these sequences.

Findings

The cross-join graph of generalized de Bruijn sequences is connected.

Any de Bruijn cycle can be transformed into any other via cross-joins.

An algorithm for Hamiltonian paths in the cross-join graph is provided.

Abstract

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.