De Bruijn Graph Homomorphisms and Recursive De Bruijn Sequences

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.

💡 Research Summary

The paper introduces a systematic method for constructing new De Bruijn cycles of higher order by exploiting cycles of lower order through specially designed homomorphisms between De Bruijn digraphs. The authors begin by reviewing the importance of De Bruijn sequences in fields such as cryptography, DNA sequencing, and robotic path planning, and they point out that the classic Lempel D‑mapping only works between consecutive orders (n and n + 1). To overcome this limitation, they define a family of “multi‑D‑morphisms” φ that map vertices of B(k, m) (the De Bruijn digraph with alphabet size k and dimension m) onto a set of k^{n‑m} vertices in B(k, n) for any n > m. The mapping is simple: each length‑m word x is extended by all possible suffixes of length n‑m, producing the words x·a₁, x·a₂, …, x·a_{k^{n‑m}}. Because φ preserves both vertices and directed edges, it is a graph homomorphism and, under the conditions stated, a graph isomorphism on the induced subgraph formed by the image of a cycle.

The core insight is that applying φ to a De Bruijn cycle C in B(k, m) yields k^{n‑m} vertex‑disjoint cycles in B(k, n). These cycles are completely independent, yet they share a common structural pattern inherited from C. The authors then describe a “cycle‑joining” procedure: for each pair of consecutive cycles in the φ‑image, a bridge edge—already present in the digraph because φ respects adjacency—is selected to connect the tail of one cycle to the head of the next. By traversing all bridges in a cyclic order, the collection of disjoint cycles collapses into a single Hamiltonian cycle that visits every vertex of B(k, n) exactly once, i.e., a new De Bruijn sequence of order n.

Mathematically, the paper proves three main theorems. Theorem 1 establishes necessary and sufficient conditions for φ to be a homomorphism that preserves adjacency (vertex‑preserving and edge‑preserving). Theorem 2 shows that under these conditions, the image of any De Bruijn cycle decomposes into exactly k^{n‑m} disjoint cycles. Theorem 3 demonstrates that the bridge‑edge construction yields a Hamiltonian cycle without creating vertex repetitions or breaking the De Bruijn property.

Building on these theoretical results, the authors present a recursive algorithm. The base case is a trivial De Bruijn cycle of order 1 (the alphabet itself). At each recursion step, the algorithm applies φ to the current cycle, generating k^{Δ} copies where Δ is the desired increase in order, and then inserts the appropriate bridge edges to merge them. The algorithm runs in O(k^{n}) time and uses O(k^{n}) memory, which is optimal because any representation of a De Bruijn sequence of order n must contain k^{n} symbols. Importantly, the method works for any alphabet size k ≥ 2, extending the applicability beyond binary sequences.

Experimental validation is performed for k = 3, 4, 5 and orders n ranging from 4 to 7. The implementation produces thousands of previously unknown De Bruijn cycles, many of which exhibit non‑canonical patterns not generated by earlier constructions. Performance measurements confirm linear growth in runtime and memory with respect to k^{n}, and the algorithm scales well with parallel execution because each copy generated by φ can be processed independently before the final joining step.

In conclusion, the paper contributes a novel framework that generalizes Lempel’s D‑mapping to arbitrary order gaps, provides a clear graph‑theoretic interpretation of cycle decomposition and reconnection, and delivers an efficient recursive construction that yields an exponential family of non‑binary De Bruijn sequences. Future work suggested includes exploring non‑linear homomorphisms, analyzing statistical properties (uniformity, autocorrelation) of the generated sequences, and applying the technique to practical problems such as pseudo‑random number generation and combinatorial testing.