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

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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References

  1. [1]

    J. Balogh, M. Kochol, A. Pluhár and X. Yu: Covering planar graphs with forests, J. Combin. Theory Ser. B 94 (2005), 147–158.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    P. A. Catlin, J. W. Grossman, A. H. Hobbs and H.-J. Lai: Fractional arboricity, strengh and principal partitions in graphs and matroids, Discrete Math. Appl. 40 (1992), 285–302.

    Article  MATH  Google Scholar 

  3. [3]

    M. Chen, S. Kim, A. Kostochka, D. B. West and X. Zhu: Decomposition of sparse graphs into forests: the Nine Dragon Tree Conjecture for k≤2, preprint, 2014.

    Google Scholar 

  4. [4]

    G. Fan, Y. Li, N. Song and D. Yang: Decomposing a graph into pseudoforests with one having bounded degree, J. Combin. Theory Ser. B 115 (2015), 72–95.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    S. L. Hakimi: On the degree of the vertices of a directed graph, J. Franklin Inst. 279 (1965), 290–308.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    D. Gonçalves: Étude de diffèrents problemes de partition de graphes, thesis, Universitè Bordeaux 1, 2006.

    Google Scholar 

  7. [7]

    D. Gonçalves: Covering planar graphs with forests, one having bounded maximum degree, J. Combin. Theory Ser. B 99 (2009), 314–322.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    S.-J. Kim, A. V. Kostochka, D. B. West, H. Wu and X. Zhu: Decomposition of sparse graphs into forests and a graph with bounded degree, J. Graph Theory 74 (2013), 369–391.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    M. Montassier, P. Ossona de Mendez, A. Raspaud and X. Zhu: Decomposing a graph into forests, J. Combin. Theory Ser. B 102 (2012), 38–52.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    C. St. J. A. Nash-Williams: Decompositions of finite graphs into forests, J. London Math. Soc. 39 (1964), 12.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    C. Payan: Graphes equilibre et arboricitè rationnelle, European J. Combin. 7 (1986), 263–270.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    J.-C. Picard and M. Queyranne: A network ow solution to some nonlinear 0-1 programming problems, with applications to graph theory, Networks 12 (1982), 141–159.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    D. Yang: Decompose a graph into forests and a matching, manuscript, 2014.

    Google Scholar 

  14. [14]

    X. Zhu: Game coloring number of planar graphs, J. Combin. Theory Ser. B 75 (1999), 245–258.

    MathSciNet  Article  MATH  Google Scholar 

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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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Key words and phrases

  • decomposition of a graph
  • arboricity
  • fractional arboricity
  • Nash-Williams’ Theorem
  • Nine Dragon Tree (NDT) Conjecture

Mathematics Subject Classification (2000)

  • 05C05
  • 05C70