Optimal packings of Hamilton cycles in graphs of high minimum degree

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We study the number of edge-disjoint Hamilton cycles one can guarantee in a sufficiently large graph G on n vertices with minimum degree d = (1/2+a)n. For any constant a > 0, we give an optimal answer in the following sense: let regₑven(n,d) denote the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree d. Then the number of edge-disjoint Hamilton cycles we find equals regₑven(n,d)/2. The value of regₑven(n,d) is known for infinitely many values of n and d. We also extend our results to graphs G of minimum degree d >= n/2, unless G is close to the extremal constructions for Dirac's theorem. Our proof relies on a recent and very general result of Kühn and Osthus on Hamilton decomposition of robustly expanding regular graphs.