Optimal factorizations of families of trees Original Research Article

Let {T1,…,Tk} be a set of trees which is Kh-packable. It is shown that every n-vertex graph G=(V,E) with δ(G)⩾n/2+3hn log n has k subgraphs S1,…,Sk with the following properties: 1. Si is a set of ⌊n/h⌋ vertex-disjoint copies of Ti. 2. The subgraphs S1,…,Sk are edge-disjoint. 3. S1∪⋯∪Sk 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+3hn log n, then G has an H-factor. • If h divides n, and δ(G)⩾n/2+3hn 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.