Optimal factorizations of families of trees

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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+3h√n 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 + 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.

Original languageEnglish
Pages (from-to)291-297
Number of pages7
JournalDiscrete Mathematics
Issue number1-3
StatePublished - 28 May 1999

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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