Factorizations of regular graphs of infinite degree

Abstract

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.