It is proven that for any integer $g\ge 0$ and $k\in \{0,\dots ,10\}$, there exist infinitely many 5-regular graphs of genus g containing a 1-factorisation with exactly k pairs of 1-factors that are perfect, i.e. form a hamiltonian cycle. For $g=0$ and $k=10$, this settles a problem of Kotzig from 1964. Motivated by Kotzig and Labelle's “marriage” operation, we discuss two gluing techniques aimed at producing graphs of high cyclic edge-connectivity. We prove that there exist infinitely many planar 5-connected 5-regular graphs in which every 1-factorisation has zero perfect pairs. On the other hand, by the Four Colour Theorem and a result of Brinkmann and the first author, every planar 4-connected 5-regular graph satisfying a condition on its hamiltonian cycles has a linear number of 1-factorisations each containing at least one perfect pair. We also prove that every planar 5-connected 5-regular graph satisfying a stronger condition contains a 1-factorisation with at most nine perfect pairs, whence, every such graph admitting a 1-factorisation with ten perfect pairs has at least two edge-Kempe equivalence classes.

证明对于任意整数 $G\ge 0$ 和 $ķ\in \{0,\dots ,10\}$，存在无限多个包含 1-因子分解的属g的 5-正则图，其中恰好有k对 1-因子是完美的，即形成一个哈密顿循环。为了$G=0$ 和 $ķ=10$，这解决了 Kotzig 从 1964 年开始的问题。受 Kotzig 和 Labelle 的“婚姻”操作的启发，我们讨论了两种旨在生成高循环边连接图的粘合技术。我们证明存在无限多个平面 5 连通 5 正则图，其中每个 1 因子分解都有零个完美对。另一方面，根据四色定理以及 Brinkmann 和第一作者的结果，每个满足其汉密尔顿循环条件的平面 4 连通 5 正则图都有线性数量的 1 因子分解，每个 1 因子分解包含至少一个完美一对。我们还证明了满足更强条件的每个平面 5 连通 5 正则图都包含一个最多包含 9 个完美对的 1 因子分解，因此，