Factorizations of Regular Graphs of Infinite Degree

Abstract

Let $\mathcal{H}=(H_i\colon i<\alpha )$ for some ordinal number $\alpha$ be an indexed family of graphs. A family $\mathcal{G}=(G_i\colon i<\alpha )$ of edge-disjoint subgraphs of a graph $G$ such that for every $i<\alpha$: $G_i$ is isomorphic to $H_i$, each $G_i$ is a spanning subgraph of $G$, and $E(G)=\bigcup\{E(G_i)\colon i < \alpha\}$ is a $\mathcal{H}$-factorization of $G$. Let $\kappa$ be an infinite cardinal. Kőnig proved in 1936 that every $\kappa$-regular graph has a factorization into perfect matchings. We extend this result to the most general factorizations possible. We study indexed families $\mathcal{T}=(T_i\colon i<\kappa)$ of graphs without isolated vertices such that every connected $\kappa$-regular graph has a $\mathcal{T}$-factorization. We prove that if $\mathcal{T}$ is a family of forests each of order at most $\kappa$, then every connected $\kappa$-regular graph $G$ has a $\mathcal{T}$-factorization. These are the most general assumptions for a family $\mathcal{T}$ for such a statement to hold.