Edge‐decomposing graphs into coprime forests
Résumé
The Barát‐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 [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.
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