Particular cycles of a binary de Bruijn digraph

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