|Résumé||The Barat-Thomassen conjecture asserts that, for every fixed tree T with t edges, there is a positive constant c_T such that every c_T-edge-connected graph G with number of edges divisible by t has a partition of its edges into copies of T. This conjecture was mainly verified for T being a tree of small diameter, until Botler, Mota, Oshiro and Wakabayashi recently proved it for T being a path of any length.
We here consider the influence of the minimum degree parameter on these considerations. As a somewhat stronger result, we prove that every 24-edge-connected graph G can be decomposed into copies of any given path, provided the path length divides |E(G)| and the minimum degree of G is large enough. Here 24 should not be optimal, as we expect the same result to hold for 3-edge-connected graphs.
This is an ongoing work with A. Harutyunyan and S. Thomassé. |