Unoriented De Bruijn Sequences

• Unoriented de Bruijn sequences are sequences over a k-symbol alphabet that include every reflected equivalence class of length-n words exactly once.
• They are constructed using the unoriented de Bruijn graph, where vertices represent reflected (n–1)-word pairs and alternating Eulerian paths ensure optimal coverage.
• Optimal sequence existence hinges on vertex parity conditions, with specific criteria satisfied for small k or n as demonstrated in recent combinatorial studies.

An unoriented de Bruijn sequence, also known as a CR-invariant or direction-blind de Bruijn sequence, is a sequence over a $k$-symbol alphabet in which each reflected equivalence class $[v]$ of length-$n$ words appears exactly once as a length-$n$ contiguous subword when the sequence is read both forwards and backwards. This property distinguishes unoriented sequences from classical de Bruijn sequences, which list all words of length $n$ only in a single orientation. The concept is defined formally as a minimal-length sequence $W$ such that for every reflected pair $[v] = \{v, v'\}$, with $v'$ denoting the reversal of $v$, exactly one member of the pair appears as a contiguous subword in $W$ or its reversal. Unoriented de Bruijn sequences arise naturally in combinatorial design, coding theory, and theoretical computer science, particularly in contexts where orientation or directionality is irrelevant or ambiguous (Burris et al., 2016).

1. Reflected Pairs and Sequence Structure

Let $[v]$0 be the $[v]$1-letter alphabet. For any $[v]$2, its reflection is defined by $[v]$3. The equivalence class $[v]$4 comprises the pair $[v]$5, with palindromic words forming singleton classes. The unoriented de Bruijn sequence $[v]$6 ensures that, for each distinct $[v]$7 of length $[v]$8, exactly one of $[v]$9 or $n$0 appears as a subword in $n$1 (possibly in $n$2 or its reverse), with palindromic representatives appearing only once.

The minimal length $n$3 of such a sequence can be explicitly computed as:

$n$4

This formula captures the count of reflected pairs plus the overlap needed at the sequence ends.

2. The Unoriented de Bruijn Graph

The central combinatorial tool is the unoriented de Bruijn graph, $n$5, an undirected multigraph where:

• Vertices: Each vertex is labeled by a reflected pair $n$6 of length-$n$7 words ($n$8).
• Edges: Each edge is labeled by a reflected pair $n$9 of length-$n$0 words ($n$1).

An edge $n$2 connects the vertices $n$3 and $n$4, where $n$5 and $n$6 are the first and last $n$7 symbols of $n$8, respectively. Loops arise when the prefix and suffix represent the same reflected pair, which happens precisely when one is the reversal of the other.

For a rigorous tracking of information, each edge incidence at a non-palindromic vertex is classified as either Type I (prefix or its reflected pair) or Type II (suffix or reflected pair).

3. Existence and Optimality Criteria

An unoriented de Bruijn sequence of optimal length exists if and only if the $n$9 admits an alternating Eulerian path, i.e., an Eulerian path alternating at each non-palindromic vertex between Type I and Type II incidences. The condition is satisfied precisely when the number of odd-degree vertices in $n$0, denoted $n$1, is at most two, with additional local parity constraints for alternation.

Key results (Burris et al., 2016):

• For $n$2 or $n$3 odd and $n$4, $n$5, and hence unoriented de Bruijn sequences of optimal length exist.
• For $n$6 even with $n$7 or $n$8, more than two odd-degree vertices occur, precluding optimal alternating Eulerian paths without further modifications.

Explicitly, for $n$9 odd and $W$0 even,

$W$1

which is only at most two for small parameter values. This parity argument governs existence of optimal sequences.

4. Algorithmic Construction via Alternating Eulerian