Proof of the 1-factorization and Hamilton decomposition conjectures

In this paper the authors prove the following results (via a unified approach) for all sufficiently large n: (i) [1-factorization conjecture] Suppose that n is even and D\geq 2\lceil n/4\rceil -1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, \chi'(G)=D. (ii) [Hamilton decomposition conjecture] Suppose that D \ge \lfloor n/2 \rfloor . Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree \delta\ge n/2. Then G contains at least {\rm reg}_{\rm even}(n,\delta)/2 \ge (n-2)/8 edge-disjoint Hamilton cycles. Here {\rm reg}_{\rm even}(n,\delta) denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree \delta. (i) was first explicitly stated by Chetwynd and Hilton. (ii) and the special case \delta= \lceil n/2 \rceil of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.