Abstract
We study the Decomposition Conjecture posed by Barát and Thomassen (2006), which states that, for each tree T, there exists a natural number kT such that, if G is a kT-edge-connected graph and |E(T)| divides |E(G)|, then G admits a partition of its edge set into copies of T. In a series of papers, Thomassen has verified this conjecture for stars, some bistars, paths of length 3, and paths whose length is a power of 2. In this paper we prove this conjecture for paths of any given length. Our technique is then used to prove weakenings of a conjecture of Kouider and Lonc (1999), and a conjecture of Favaron, Genest and Kouider (2010), both for path decomposition of regular graphs.
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