An Unoriented Variation on de Bruijn Sequences

For positive integers k, n, a de Bruijn sequence B(k, n) is a finite sequence of elements drawn from k characters whose subwords of length n are exactly the \(k^n\) words of length n on k characters. This paper studies the unoriented de Bruijn sequence uB(k, n), an analog to de Bruijn sequences, but for which the sequence is read both forwards and backwards to determine the set of subwords of length n. We show that nontrivial unoriented de Bruijn sequences of optimal length exist if and only if k is two or odd and n is less than or equal to 3. Unoriented de Bruijn sequences for any k, n may be constructed from certain Eulerian trails in Eulerizations of unoriented de Bruijn graphs.