Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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.

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The categories of this paper are: supr-con
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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an