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

Abstract

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.

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Correspondence to Julien Bensmail.

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The first author was supported by ERC Advanced Grant GRACOL, project no. 320812. The second author was supported by an FQRNT postdoctoral research grant and CIMI research fellowship. The fourth author was partially supported by the ANR Project STINT under Contract ANR-13-BS02-0007.

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Bensmail, J., Harutyunyan, A., Le, T. et al. Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture. Combinatorica 39, 239–263 (2019). https://doi.org/10.1007/s00493-017-3661-5

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Mathematics Subject Classification (2010)

  • 05C40
  • 05C07
  • 05C15
  • 05C38