We prove that for any pair of constants a>0 and D and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most D, and with at most n(n-1)/2 edges in total packs into the complete graph of order (1+a)n. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.
@InProceedings{bottcher_et_al:LIPIcs.APPROX-RANDOM.2014.490, author = {B\"{o}ttcher, Julia and Hladk\'{y}, Jan and Piguet, Diana and Taraz, Anusch}, title = {{An Approximate Version of the Tree Packing Conjecture via Random Embeddings}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {490--499}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.490}, URN = {urn:nbn:de:0030-drops-47184}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.490}, annote = {Keywords: tree packing conjecture, Ringel’s conjecture, random walks, quasirandom graphs} }
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