De Bruijn Graph Homomorphisms and Recursive De Bruijn Sequences

TL;DR

This paper introduces a method for constructing new De Bruijn cycles of higher order by mapping existing cycles through homomorphisms, generalizing known morphisms and providing an efficient recursive algorithm for generating numerous cycles.

Contribution

It characterizes homomorphisms between De Bruijn digraphs of different orders and develops a recursive algorithm to generate many nonbinary De Bruijn cycles.

Findings

Characterization of homomorphisms between De Bruijn digraphs of different orders.

A recursive algorithm for generating exponential numbers of nonbinary De Bruijn cycles.

Generalization of the D-morphism of Lempel to higher-order De Bruijn digraphs.

Abstract

This paper presents a method to find new De Bruijn cycles based on ones of lesser order. This is done by mapping a De Bruijn cycle to several vertex disjoint cycles in a De Bruijn digraph of higher order and connecting these cycles into one full cycle. We characterize homomorphisms between De Bruijn digraphs of different orders that allow this construction. These maps generalize the well-known D-morphism of Lempel between De Bruijn digraphs of consecutive orders. Also, an efficient recursive algorithm that yields an exponential number of nonbinary De Bruijn cycles is implemented.