  A decomposition of a multigraph $G$ is a partition of its edges into
subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every
$G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a
decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for
every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$.
  Feghali and Johnson gave necessary and sufficient conditions for a given
decomposition of $\lambda K_n$ to be enclosed in some $2$-edge-connected
$r$-factorization of $\mu K_{m}$ for some range of values for the parameters
$n$, $m$, $\lambda$, $\mu$, $r$: $r=2$, $\mu>\lambda$ and either $m \geq 2n-1$,
or $m=2n-2$ and $\mu = 2$ and $\lambda=1$, or $n=3$ and $m=4$. We generalize
their result to every $r \geq 2$ and $m \geq 2n - 2$. We also give some
sufficient conditions for enclosing a given decomposition of $\lambda K_n$ in
some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for every $r \geq 3$
and $m = (2 - C)n$, where $C$ is a constant that depends only on $r$, $\lambda$
and~$\mu$.
