  Let $n\ge 34$ be an even integer, and $D_n=2\lceil n/4 \rceil-1$. In this
paper, we prove that every $\{D_n,\,D_n+1\}$-graph of order $n$ contains
$\lceil n/4 \rceil$ disjoint perfect matchings. This result is sharp in the
sense that (i) there exists a $\{D_n,\,D_n+1\}$-graph containing exactly
$\lceil n/4 \rceil$ disjoint perfect matchings, and that (ii) there exists a
$\{D_n-1,\,D_n\}$-graph without perfect matchings for each $n$. As a
consequence, for any integer $D\ge D_n$, every $\{D,\,D+1\}$-graph of order $n$
contains $\lceil (D+1)/2 \rceil$ disjoint perfect matchings. This extends Csaba
et~al.'s breathe-taking result that every $D$-regular graph of sufficiently
large order is $1$-factorizable, generalizes Zhang and Zhu's result that every
$D_n$-regular graph of order $n$ contains $\lceil n/4 \rceil$ disjoint perfect
matchings, and improves Hou's result that for all $k\ge n/2$, every
$\{k,\,k+1\}$-graph of order $n$ contains $(\lfloor n/3\rfloor+1+k-n/2)$
disjoint perfect matchings.
