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, 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.
Proof of the 1-factorization and Hamilton Decomposition Conjectures (Memoirs of the American Mathematical Society) pdf
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