Optimal packings of bounded degree trees
We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gy\'arf\'as and Lehel from 1976 holds for all bounded degree trees (in fact, we can a...
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| Hauptverfasser: | , , , |
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| Dokumenttyp: | Article (Journal) Kapitel/Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
13 Jun 2016
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| In: |
Arxiv
Year: 2016, Pages: 1-56 |
| DOI: | 10.48550/arXiv.1606.03953 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.1606.03953 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/1606.03953 
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| Verfasserangaben: | Felix Joos, Jaehoon Kim, Daniela Kühn, and Deryk Osthus |
| Zusammenfassung: | We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gy\'arf\'as 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 Szemer\'{e}di'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. |
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| Beschreibung: | Identifizierung der Ressource nach: 13 Mar 2019 Gesehen am 27.07.2022 |
| Beschreibung: | Online Resource |
| DOI: | 10.48550/arXiv.1606.03953 |