Abstract
We prove the following results (via a unified approach) for all sufficiently large n:
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(i)
[1 -factorization conjecture] Suppose that n is even and D ≥ 2⌈n/4⌉ − 1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ′(G) = D.
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(ii)
[Hamilton decomposition conjecture] Suppose that D ≥ ⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching.
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(iii)
[Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree δ ≥ n/2. Then G contains at least (n − 2)/8 edge-disjoint Hamilton cycles. According to Dirac, (i) was first raised in the 1950’s. (ii) and (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.
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Csaba, B., Kühn, D., Lo, A., Osthus, D., Treglown, A. (2013). Proof of the 1-factorization and Hamilton decomposition conjectures. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_76
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DOI: https://doi.org/10.1007/978-88-7642-475-5_76
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