Proof of the 1-factorization and Hamilton Decomposition Conjectures

Abstract

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, 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 (n − 2)/8 edge-disjoint Hamilton cycles. According to Dirac, (i) was first raised in the 1950’s. (ii) and (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.