  A subgraph $H$ of a multigraph $G$ is overfull if $ |E(H) | > \Delta(G)
\lfloor |V(H)|/2 \rfloor$. Analogous to the Overfull Conjecture proposed by
Chetwynd and Hilton in 1986, Stiebitz et al. in 2012 formed the multigraph
version of the conjecture as follows: Let $G$ be a multigraph with maximum
multiplicity $r$ and maximum degree $\Delta>\frac{1}{3} r|V(G)|$. Then $G$ has
chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph.
In this paper, we prove the following three results toward the Multigraph
Overfull Conjecture for sufficiently large and even $n$.
  (1) If $G$ is $k$-regular with $k\ge r(n/2+18)$, then $G$ has a
1-factorization. This result also settles a conjecture of the first author and
  Tipnis from 2001 up to a constant error in the lower bound of $k$.
  (2) If $G$ contains an overfull subgraph and $\delta(G)\ge r(n/2+18)$, then
$\chi'(G)=\lceil \chi'_f(G) \rceil$, where $\chi'_f(G)$ is the fractional
chromatic index of $G$.
  (3) If the minimum degree of $G$ is at least $(1+\varepsilon)rn/2$ for any
$0<\varepsilon<1$ and $G$ contains no overfull subgraph, then
$\chi'(G)=\Delta(G)$.
  The proof is based on the decomposition of multigraphs into simple graphs and
we prove a slightly weak version of a conjecture due to the first author and
Tipnis from 1991 on decomposing a multigraph into constrained simple graphs.
The result is of independent interests.
