LIPIcs.APPROX-RANDOM.2014.490.pdf
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An Approximate Version of the Tree Packing Conjecture via Random Embeddings
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Randomization and Computation (RANDOM)
Conference: International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) - License:
Creative Commons Attribution 3.0 Unported license
- Publication date: 2014-09-04
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Julia Böttcher, Jan Hladký, Diana Piguet, and Anusch Taraz. An Approximate Version of the Tree Packing Conjecture via Random Embeddings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 490-499, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.490
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.490
Abstract
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.
Keywords
- tree packing conjecture
- Ringel’s conjecture
- random walks
- quasirandom graphs
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