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An approximate version of the tree packing conjecture

Israel Journal of Mathematics volume 211, pages 391–446 (2016)Cite this article

Abstract

We prove that for any pair of constants ɛ > 0 and Δ and for n sufficiently large, every family of trees of orders at most n, maximum degrees at most Δ, and with at most ( 2 n ) edges in total packs into \({K_{(1 + \varepsilon )n}}\). This implies asymptotic versions of the Tree Packing Conjecture of Gyárfás 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.

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Authors and Affiliations

  1. Department of Mathematics, London School of Economics Houghton Street, London, WC2A 2AE, UK

    Julia Böttcher

  2. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, Praha, Czech Republic

    Jan Hladký

  3. Institute of Computer Science, Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07, Prague, Czech Republic

    Diana Piguet

  4. Institut für Mathematik, Technische Universität Hamburg-Harburg, Am Schwarzenberg-Campus 3, 21073, Hamburg, Germany

    Anusch Taraz

Authors
  1. Julia Böttcher
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  2. Jan Hladký
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  3. Diana Piguet
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  4. Anusch Taraz
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Corresponding author

Correspondence to Julia Böttcher.

Additional information

Support by the Institut Mittag-Leffler (Djursholm, Sweden) is gratefully acknowledged.

The Institute of Mathematics is supported by RVO:67985840. The work was done while the author was an EPSRC Research Fellow at DIMAP and Mathematics Institute at the University of Warwick.

The Institute of Computer Science has institutional support RVO:67985807. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. PIEF-GA-2009-253925.

The author was supported in part by DFG grant TA 319/2-2.

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Böttcher, J., Hladký, J., Piguet, D. et al. An approximate version of the tree packing conjecture. Isr. J. Math. 211, 391–446 (2016). https://doi.org/10.1007/s11856-015-1277-2

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  • DOI: https://doi.org/10.1007/s11856-015-1277-2

Keywords

  • Maximum Degree
  • Common Neighbourhood
  • Primary Vertex
  • Secondary Vertex
  • Host Graph