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

  • Julien Bensmail
  • Ararat Harutyunyan
  • Tien-Nam Le
  • Stéphan Thomasse


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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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Julien Bensmail
    • 1
  • Ararat Harutyunyan
    • 2
  • Tien-Nam Le
    • 3
  • Stéphan Thomasse
    • 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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