, Volume 39, Issue 2, pp 239–263 | Cite as

Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture

  • Julien BensmailEmail author
  • Ararat Harutyunyan
  • Tien-Nam Le
  • Stéphan Thomassé


In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f(t)-edge-connected graph with its number of edges divisible by t has a partition of its edges into copies of T. This conjecture was recently verified by the current authors and Merker [1].

We here further focus on the path case of the Barát-Thomassen conjecture. Before the aforementioned general proof was announced, several successive steps towards the path case of the conjecture were made, notably by Thomassen [11,12,13], until this particular case was totally solved by Botler, Mota, Oshiro andWakabayashi [2]. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function f such that every 24-edge-connected graph with minimum degree f(t) has an edge-partition into paths of length t whenever t divides the number of edges. We also show that 24 can be dropped to 4 when the graph is eulerian.

Mathematics Subject Classification (2010)

05C40 05C07 05C15 05C38 


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  1. [1]
    J. Bensmail, A. Harutyunyan, T.-N. Le, M. Merker and S. Thomassé: A Proof of the Barát-Thomassen Conjecture. Journal of Combinatorial Theory, Series B 124 (2017), 39–55.zbMATHGoogle Scholar
  2. [2]
    F. Botler, G. O. Mota, M. Oshiro and Y. Wakabayashi: Decomposing highly edge-connected graphs into paths of any given length. Journal of Combinatorial Theory, Series B 122 (2017), 508–542.zbMATHGoogle Scholar
  3. [3]
    F. Botler, G. O. Mota, M. Oshiro and Y. Wakabayashi: Decompositions of highly connected graphs into paths of length five. Discrete Applied Mathematics, Doi: 10.1016/j.dam.2016.08.001, 2016.Google Scholar
  4. [4]
    J. Barát and C. Thomassen: Claw-decompositions and Tutte-orientations. Journal of Graph Theory 52 (2006), 135–146.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J. Edmonds: Edge-disjoint branchings, Combinatorial Algorithms (B. Rustin, editor), 91–96, Academic Press, 1973.Google Scholar
  6. [6]
    B. Jackson: On circuit covers, circuit decompositions and Euler tours of graph, Surveys in Combinatorics, London Mathematical Society Lecture Note Series, 187 (1993), 191–210.zbMATHGoogle Scholar
  7. [7]
    C. McDiarmid: Concentration for Independent Permutations. Combinatorics, Probability and Computing 11 (2002), 163–178.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Molloy and B. Reed: Graph Colouring and the Probabilistic Method. Springer, 2002.CrossRefzbMATHGoogle Scholar
  9. [9]
    C. St. J. A. Nash-Williams: On orientations, connectivity and odd-vertex-pairings in finite graphs. Canadian Journal of Mathematics 12 (1960), 555–567.zbMATHGoogle Scholar
  10. [10]
    M. Stiebitz, D. Scheide, B. Toft and L. M. Favrholdt: Graph Edge Coloring: Vizing's Theorem and Goldberg's Conjecture, Wiley, 2012.zbMATHGoogle Scholar
  11. [11]
    C. Thomassen: Decompositions of highly connected graphs into paths of length 3. Journal of Graph Theory 58 (2008), 286–292.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    C. Thomassen: Edge-decompositions of highly connected graphs. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 18 (2008), 17–26.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    C. Thomassen: Decomposing graphs into paths of fixed length. Combinatorica 33 (2013), 97–123.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

  • Julien Bensmail
    • 1
    Email author
  • Ararat Harutyunyan
    • 2
  • Tien-Nam Le
    • 3
  • Stéphan Thomassé
    • 3
  1. 1.I3S and INRIAUniversité Nice-Sophia-AntipolisSophia-AntipolisFrance
  2. 2.LAMSADE, CNRSUniversité Paris-Dauphine PSL Research UniversityParisFrance
  3. 3.Laboratoire d’Informatique du ParallélismeÉcole Normale Supérieure de LyonLyon Cedex 07France

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