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 $D\geq 2\lceil n/4\rceil -1$. 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 $\text{reg}_{\text{even}}(n,\delta)$ 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.