Particular cycles of a binary de Bruijn digraph
The authors investigate an isomorphism \(\sigma \) 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 \(\sigma \)-self-converse when the cycle \(\sigma (C)\) coincides with its converse. A characterization is determined of \(\sigma \)-self-converse cycles, distinguishing the cases of \(n\) even and odd. Moreover it is proved that, for \(n\) even, there does not exist a Hamiltonian \(\sigma \)-self-converse cycle, while, for \(n\) odd, we determine a constructive proof of the existence of a similar cycle. Finally they prove that for every \(n\) there exists only one \(\sigma \)-self-converse cycle of length 4.
scientific article; zbMATH DE number 5702102