Particular cycles of a binary de Bruijn digraph

In this paper we investigate an isomorphism σ between a directed de Bruijn digraph B ( 2 , n ) and its converse, which is the digraph obtained from B ( 2 , n ) by reversing the direction of all its arcs. A cycle C is said to be σ -self-converse when the cycle σ ( C ) coincides with its converse. We determine a characterization of σ -self-converse cycles, distinguishing the cases of n even and odd. Moreover we prove that, for n even, there does not exist a Hamiltonian σ -self-converse cycle, while, for n odd, we determine a constructive proof of the existence of a similar cycle. Finally we prove that for every n there exists only one σ -self-converse cycle of length 4.