Factorizing the Rado graph and infinite complete graphs

Abstract

Let $\mathcal{F}=\{F_α: α\in \mathcal{A}\}$ be a family of infinite graphs, together with $Λ$. The Factorization Problem $FP(\mathcal{F}, Λ)$ asks whether $\mathcal{F}$ can be realized as a factorization of $Λ$, namely, whether there is a factorization $\mathcal{G}=\{Γ_α: α\in \mathcal{A}\}$ of $Λ$ such that each $Γ_α$ is a copy of $F_α$. We study this problem when $Λ$ 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.