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Edge-partitioning a graph into paths: beyond the Barát-Thomassen conjecture

Julien Bensmail 1 Ararat Harutyunyan 2 Tien-Nam Le 3 Stéphan Thomassé 4, 5
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
4 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
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. 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, until this particular case was totally solved by Botler, Mota, Oshiro and Wakabayashi. 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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Julien Bensmail, Ararat Harutyunyan, Tien-Nam Le, Stéphan Thomassé. Edge-partitioning a graph into paths: beyond the Barát-Thomassen conjecture. Combinatorica, Springer Verlag, 2019, 39 (2), pp.239-263. ⟨10.1007/s00493-017-3661-5⟩. ⟨hal-01744515⟩

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