Proof of the 1-factorization and Hamilton decomposition conjectures II: the bipartite case

Understand

In a sequence of four papers, 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 $.