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 .
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 . 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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J. Bensmail, A. Harutyunyan, T.-N. Le, M. Merker and S. Thomassé: A Proof of the Barát-Thomassen Conjecture. Journal of Combinatorial Theory, Series B124 (2017), 39–55.zbMATHGoogle Scholar
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 B122 (2017), 508–542.zbMATHGoogle Scholar
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