Edge-decomposing graphs into coprime forests

The Barát-Thomassen conjecture, recently proved in [3], asserts that for every tree T , there is a constant cT such that every cT -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 [4] that when T is a path with k edges, there is a constant dk such that every 24-edge connected graph G with size divisible by k and minimum degree dk 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 [4] 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. ∗Both authors were partially supported by ANR project Stint under reference ANR-13BS02-0007 and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program Investissements d’Avenir (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Klimošová was also supported by Center of Excellence – ITI, project P202/12/G061 of GA ČR and by Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004). †Extended abstract of this work was published as: T. Klimošová, S. Thomassé: Decomposing graphs into paths and trees, Electron. Notes Discrete Math., 61 (2017) 751-757. ‡Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské náměst́ı 25, 118 00 Praha 1, Czech Republic. E-mail: [email protected]. §Laboratoire d’Informatique du Parallélisme, École Normale Supérieure de Lyon, 69364 Lyon Cedex 07, France. E-mail: [email protected]. ¶Institut Universitaire de France 1 ar X iv :1 80 3. 03 70 4v 1 [ m at h. C O ] 9 M ar 2 01 8