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Decomposing a Graph into Forests: The Nine Dragon Tree Conjecture is True

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Abstract

The fractional arboricity of a graph G, denoted by Γf (G), is defined as

$${\Gamma _f}\left( G \right) = {\max _{H \subseteq G,v\left( H \right) > 1}}\frac{{e\left( H \right)}}{{v\left( H \right) - 1}}$$

. The celebrated Nash-Williams’ Theorem states that a graph G can be partitioned into at most k forests if and only if Γf (G)≤k. The Nine Dragon Tree (NDT) Conjecture [posed by Montassier, Ossona de Mendez, Raspaud, and Zhu, in “Decomposing a graph into forests”, J. Combin. Theory Ser. B 102 (2012) 38-52] asserts that if

$${\Gamma _f}\left( G \right) \leqslant k + \frac{d}{{k + d + 1}}$$

, then G decomposes into k+1 forests with one having maximum degree at most d. In this paper, we prove the Nine Dragon Tree (NDT) Conjecture.

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Authors and Affiliations

  1. Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian, 350003, China

    Hongbi Jiang & Daqing Yang

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  1. Hongbi Jiang
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  2. Daqing Yang
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Correspondence to Daqing Yang.

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Supported in part by NSFC under grant 11471076.

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Jiang, H., Yang, D. Decomposing a Graph into Forests: The Nine Dragon Tree Conjecture is True. Combinatorica 37, 1125–1137 (2017). https://doi.org/10.1007/s00493-016-3390-1

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  • DOI: https://doi.org/10.1007/s00493-016-3390-1

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