Generalized Orthogonal de Bruijn and Kautz Sequences

TL;DR

This work introduces and analyzes generalized orthogonal de Bruijn and Kautz sequences by relaxing or constraining the substring coverage and weight constraints. It develops $\,\ell$-orthogonality, $b$-balanced, and fixed-weight variants, deriving tight or near-tight bounds on the maximum number of pairwise compatible sequences through Eulerian/Hamiltonian graph characterizations, vertex-splitting, wirings, and tensor-product constructions. The results extend to nonbinary Kautz sequences with parallel theorems and constructive methods, providing both theoretical bounds and practical construction techniques. The findings have implications for synthesis-constrained sequence design in applications such as DNA-based data storage and probe design, where runlength and composition constraints are critical.

Abstract

A de Bruijn sequence of order $k$ over a finite alphabet is a cyclic sequence with the property that it contains every possible $k$-sequence as a substring exactly once. Orthogonal de Bruijn sequences are collections of de Bruijn sequences of the same order, $k$, satisfying the joint constraint that every $(k+1)$-sequence appears as a substring in at most one of the sequences in the collection. Both de Bruijn and orthogonal de Bruijn sequences have found numerous applications in synthetic biology, although the latter remain largely unexplored in the coding theory literature. Here we study three relevant practical generalizations of orthogonal de Bruijn sequences where we relax either the constraint that every $(k+1)$-sequence appears exactly once, or that the sequences themselves are de Bruijn rather than balanced de Bruijn sequences. We also provide lower and upper bounds on the number of fixed-weight orthogonal de Bruijn sequences. The paper concludes with parallel results for orthogonal nonbinary Kautz sequences, which satisfy similar constraints as de Bruijn sequences except for only being required to cover all subsequences of length $k$ whose maximum runlength equals to one.

Figures (5)

• Figure 1: \ref{['subfig:dB_3_2_original']} The de Bruijn graph $G_{3,2}$. \ref{['subfig:dB_3_2_012002211']} An Eulerian circuit in $G_{3,2}$ that represents the $(3,2)$-de Bruijn sequence $012002211$. \ref{['subfig:dB_3_3_012002211']} The de Bruijn graph $G_{3,3}$. The arcs in the Hamiltonian cycle of the sequence $012002211$ are depicted with bold lines.
• Figure 2: \ref{['subfig:rewiring_C_1_2']} The circuit $C_{1,2}=012022110$ obtained by rewiring $C_{1,1}=012002211$ at the vertex $0$, given $C_{1,1}$. \ref{['subfig:rewiring_C_2_1']} The circuit $C_{2,1}=011220210$ obtained by rewiring $C_{1,2}$ at the vertices $1$ and $2$, given $C_{1,1}$. \ref{['subfig:rewiring_C_2_2']} The circuit $C_{2,2}=011220021$ obtained by rewiring $C_{2,1}$ at the vertex $0$, given $C_{2,1}$.
• Figure 3: The de Bruijn graph $G_{3,3}$. The bold arcs correspond to the $2$-circuit representing the $2$-balanced $(3,2)$-de Bruijn sequence $002211012001122021$.
• Figure 4: \ref{['subfig:Kautz_4_2']} The Kautz graph $G^{\textnormal{Kautz}}_{4,2}$. \ref{['subfig:Kautz_4_2_wired']} The wiring of $G^{\textnormal{Kautz}}_{4,2}$ induced by the Eulerian circuit corresponding to the $(4,2)$-Kautz sequence ATCGAGCTGTAC. \ref{['subfig:Kautz_4_3']} The Kautz graph $G^{\textnormal{Kautz}}_{4,3}$.
• Figure 5: Some fixed-weight Kautz graphs with alphabet $\mathcal{A}=\{A,T,C,G\}$ and $k=3$. \ref{['subfig:Kautz_fixed_wgt_4_3_1_1']} The fixed-weight Kautz graph $D(\mathcal{K}_3(\mathcal{A})\cap\mathcal{A}_{1}^1(3))$. \ref{['subfig:Kautz_fixed_wgt_4_3_1_2']} The fixed-weight Kautz graph $D(\mathcal{K}_3(\mathcal{A})\cap\mathcal{A}_{1}^2(3))$.

Theorems & Definitions (37)

• Definition 1
• Definition 2
• Example 1
• Definition 3
• Definition 4
• Proposition 1
• Theorem 1
• Definition 5
• Lemma 1
• Lemma 2