It is proven that for any integer $g \ge 0$ and $k \in \{ 0, \ldots, 10 \}$, there exist infinitely many 5-regular graphs of genus $g$ containing a 1-factorisation with exactly $k$ pairs 1-factors are perfect, i.e. form hamiltonian cycle. For = 0$, this settles problem Kotzig from 1964. Motivated by Labelle's "marriage" operation, we discuss two gluing techniques aimed at producing high cyclic ...