  Let $\mathcal{H}=\{H_i: i<\alpha \}$ be an indexed family of graphs for some
ordinal number $\alpha$. $\mathcal{H}$-decomposition of a graph $G$ is a family
$\mathcal{G}=\{G_i: i<\alpha \}$ of edge-disjoint subgraphs of $G$ such that
$G_i$ is isomorphic to $H_i$ for every $i<\alpha$ and
$\bigcup\{E(G_i):i<\alpha\}=E(G)$. $\mathcal{H}$-factorization of $G$ is a
$\mathcal{H}$-decomposition of $G$ such that every element of $\mathcal{H}$ is
a spanning subgraph of $G$. Let $\kappa$ be an infinite cardinal. K\H{o}nig in
1936 proved that every $\kappa$-regular graph has a factorization into perfect
matchings. Andersen and Thomassen using this theorem proved in 1980 that every
$\kappa$-regular connected graph has a $\kappa$-regular spanning tree. We
generalize both these results and establish the existence of a factorization of
$\kappa$-regular graph into $\lambda$-regular subgraphs for every non-zero
$\lambda\leq \kappa$. Furthermore, we show that every $\kappa$-regular
connected graph has a $\mathcal{H}$-factorization for every family
$\mathcal{H}$ of $\kappa$ forests with $\kappa$ components of order at most
$\kappa$ and without isolated vertices.
