Perfect matchings of a graph associated with a binary de Bruijn digraph

Let BnBn be the binary de Bruijn digraph of order nn and WW the quotient set of the set of vertices of BnBn with respect to the equivalence relation of rotation. Let GG be the graph which has WW as the set of vertices and in which two elements CC and HH are adjacent when there exist a vertex vv of CC and a vertex uu of HH such that (v,u)(v,u) is an arc of BnBn. Recently the problem of establishing whether the graph GG has a perfect matching was posed. In this paper we answer in the positive to this problem in the case of nn odd.