Factorizing the Rado graph and infinite complete graphs

TL;DR

This paper investigates the factorization problem for infinite graphs, specifically the Rado graph and infinite complete graphs, providing conditions for when such factorizations exist and exploring specific cases like countable stars.

Contribution

It characterizes when the factorization problem has solutions for the Rado graph and infinite complete graphs, including new non-existence and existence results for certain graph decompositions.

Findings

Factorization of the Rado graph is possible if and only if graphs have no finite dominating set.

Complete graph factorizations exist when graphs match the order and domination numbers.

No factorization into 1- or 2-star decompositions of countable complete graphs; exists for 4-stars, open for 3-stars.

Abstract

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.