Proof of the 1 factorization and Hamilton Decomposition Conjectures

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