Proof of the 1-Factorization and Hamilton Decomposition Conjectures

In this paper we prove the following results (via a unified approach) for all sufficiently large n: (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, chi' (G) = D. (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. (iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree delta >= n/2. Then G contains at least reg(even)(n, delta)/2 >= (n- 2)/8 edge-disjoint Hamilton cycles. Here reg(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 = [n/2] of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.