On Tree-Partition-Width

Abstract

A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag in a tree-partition of $G$. An anonymous referee of the paper by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph with tree-width $k\geq3$ and maximum degree $Δ\geq1$ has tree-partition-width at most $24kΔ$. We prove that this bound is within a constant factor of optimal. In particular, for all $k\geq3$ and for all sufficiently large $Δ$, we construct a graph with tree-width $k$, maximum degree $Δ$, and tree-partition-width at least $(\eighth-ε)kΔ$. Moreover, we slightly improve the upper bound to ${5/2}(k+1)({7/2}Δ-1)$ without the restriction that $k\geq3$.