Factorizations of regular graphs of infinite degree

TL;DR

This paper generalizes classical results on factorization of infinite regular graphs, proving they can be decomposed into regular subgraphs of smaller degrees and into forests with specific properties.

Contribution

It extends Kőnig's and Andersen-Thomassen's theorems by establishing factorizations into -regular subgraphs and -forest factorizations for infinite regular graphs.

Findings

Every -regular graph can be factored into -regular subgraphs for all    .

Every -regular connected graph admits a -forest -factorization.

Generalizes classical theorems to broader classes of infinite regular graphs.

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.