Particular cycles of a binary de Bruijn digraph

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