Abstrakt
The Barat-Thomassen conjecture, recently proved in Bensmail et al. (2017), asserts that for every tree T, there is a constant c T such that every c T-edge-connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T-decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition.
For instance, it was shown in Bensmail et al. (2019) that when T is a path with k edges, there is a constant d k such that every 24-edge-connected graph G with size divisible by k and minimum degree d k has a T-decomposition. We show in this paper that when F is a coprime forest (the sizes of its components being a coprime set of integers), any graph G with sufficiently large minimum degree has an F-decomposition provided that the size of F divides the size of G (no connectivity is required).
A natural conjecture asked in Bensmail et al. (2019) asserts that for a fixed tree T, any graph G of size divisible by the size of T with sufficiently high minimum degree has a T-decomposition, provided that G is sufficiently highly connected in terms of the maximal degree of T. The case of maximum degree 2 is answered by paths.
We provide a counterexample to this conjecture in the case of maximum degree 3.