Optimal factorizations of families of trees

Let {T₁,...,T_k} be a set of trees which is K_h-packable. It is shown that every n-vertex graph G=(V,E) with δ(G) ≥n/2+3h√n log n has k subgraphs S₁,...,S_k with the following properties: 1. S_i is a set of [n/h] vertex-disjoint copies of T_i. 2. The subgraphs S₁,...,S_k are edge-disjoint. 3. S₁ ∪ ... ∪ S_k has maximum degree at most h -1. There are many interesting special cases of this result. To name just two: • If H is a tree with h vertices and G = (V,E) is a graph with n vertices, h divides n, and δ(G)≥n/2 + 3h√n log n, then G has an H-factor. • If h divides n, and δ(G)≥n/2 + 3h√n log n, then G has a set S of n star subgraphs, where for each i= 1,...,h, there are exactly n/h stars in S having i vertices, any two members of S having the same size are vertex-disjoint, and the union of all the members of S is an h - 1 regular spanning subgraph of G.