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- W3048233797 abstract "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 of 1-factors that are perfect, i.e. form a hamiltonian cycle. For $g = 0$, 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. The paper concludes with further results on edge-Kempe equivalence classes in planar 5-regular graphs." @default.
- W3048233797 created "2020-08-13" @default.
- W3048233797 creator A5031033019 @default.
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- W3048233797 date "2020-08-07" @default.
- W3048233797 modified "2023-09-27" @default.
- W3048233797 title "Hamiltonian cycles and 1-factors in 5-regular graphs" @default.
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