On tree-partition-width

A 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 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 [J. Graph Theory, 1995] proved that every graph with tree-width k ≥ 3 and maximum degree ∆ ≥ 1 has tree-partition-width at most 24k∆. We prove that this bound is within a constant factor of optimal. In particular, for all k ≥ 3 and for all sufficiently large ∆, we construct a graph with tree-width k, maximum degree ∆, and tree-partition-width at least ( 1 8 − ǫ)k∆. Moreover, we slightly improve the upper bound to 5 2 (k + 1)( 7 2 ∆ − 1) without the restriction that k ≥ 3.