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Packing spanning graphs from separable families

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Abstract

Let \(\mathcal{G}\) be a separable family of graphs. Then for all positive constants ϵ and Δ and for every sufficiently large integer n, every sequence G 1,..., G t ∈ \(\mathcal{G}\) of graphs of order n and maximum degree at most Δ such that

$$\left( {{G_1}} \right) + \cdots + e\left( {{G_t}} \right) \leqslant \left( {1 - \epsilon } \right)\left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right)$$

packs into K n . This improves results of Böttcher, Hladký, Piguet and Taraz when \(\mathcal{G}\) is the class of trees and of Messuti, Rödl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyárfás and Lehel (1976) for the case that all trees have maximum degree at most Δ.

The proof uses the local resilience of random graphs and a special multi-stage packing procedure.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, 06520, USA

    Asaf Ferber

  2. Department of Mathematics, MIT, Cambridge, MA, 02139-4307, USA

    Asaf Ferber & Choongbum Lee

  3. Department of Computer Science, ETH Zürich, 8092, Zürich, Switzerland

    Frank Mousset

Authors
  1. Asaf Ferber
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  2. Choongbum Lee
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  3. Frank Mousset
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Corresponding author

Correspondence to Frank Mousset.

Additional information

Research supported by NSF Grant DMS-1362326.

Supported by grant no. 6910960 of the Fonds National de la Recherche, Luxembourg.

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Ferber, A., Lee, C. & Mousset, F. Packing spanning graphs from separable families. Isr. J. Math. 219, 959–982 (2017). https://doi.org/10.1007/s11856-017-1504-0

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  • DOI: https://doi.org/10.1007/s11856-017-1504-0

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