Optimal packings of bounded degree trees

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Abstract

We prove that if T1,…,Tn is a sequence of bounded degree trees such that Ti has i vertices, then Kn has a decomposition into T1,…,Tn. This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.

Details

Original languageEnglish
JournalJournal of the European Mathematical Society
Volume21
Issue number12
Publication statusPublished - 5 Aug 2019

Keywords

  • Graph decompositions, Packings, Quasirandomness, Trees

ASJC Scopus subject areas