An approximate version of the tree packing conjecture

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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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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Keywords

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