Hamiltonian cycles and 1-factors in 5-regular graphs

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 edge-connectivity. We prove planar 5-connected in which every has zero perfect pairs. On the other hand, Four Colour Theorem result Brinkmann first author, 4-connected graph satisfying condition on its cycles linear number 1-factorisations each least one pair. also stronger contains most nine pairs, whence, such admitting ten edge-Kempe equivalence classes. The paper concludes further results classes graphs.