  Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of
infinite graphs, together with $\Lambda$. The Factorization Problem
$FP(\mathcal{F}, \Lambda)$ asks whether $\mathcal{F}$ can be realized as a
factorization of $\Lambda$, namely, whether there is a factorization
$\mathcal{G}=\{\Gamma_{\alpha}: \alpha\in \mathcal{A}\}$ of $\Lambda$ such that
each $\Gamma_{\alpha}$ is a copy of $F_{\alpha}$.
  We study this problem when $\Lambda$ is either the Rado graph $R$ or the
complete graph $K_\aleph$ of infinite order $\aleph$. When $\mathcal{F}$ is a
countable family, we show that $FP(\mathcal{F}, R)$ is solvable if and only if
each graph in $\mathcal{F}$ has no finite dominating set. We also prove that
$FP(\mathcal{F}, K_\aleph)$ admits a solution whenever the cardinality
$\mathcal{F}$ coincide with the order and the domination numbers of its graphs.
  For countable complete graphs, we show some non existence results when the
domination numbers of the graphs in $\mathcal{F}$ are finite. More precisely,
we show that there is no factorization of $K_{\mathbb{N}}$ into copies of a
$k$-star (that is, the vertex disjoint union of $k$ countable stars) when
$k=1,2$, whereas it exists when $k\geq 4$, leaving the problem open for $k=3$.
  Finally, we determine sufficient conditions for the graphs of a decomposition
to be arranged into resolution classes.
