Decomposition of Graphs into Paths

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 jE(T )j divides jE(G)j, 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.