Tree-Partitions with Bounded Degree Trees

A tree-partition of a graph G is a partition of V(G) such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth k and maximum degree \(\Delta \) has a tree-partition with parts of size \(O(k\Delta )\) . We prove the same result with the extra property that the underlying tree has maximum degree \(O(\Delta )\) .