Decomposition of Graphs into Paths

A decomposition of a graph G is a set D = {H1, · · · , Hk} of pairwise edge-disjoint subgraphs of G that cover the set of edges of G. If Hi is isomorphic to a fixed graph H, for 1 ≤ i ≤ k, then we say that D is an H-decomposition of G. In this work, we study the case where H is a path of fixed length. For that, we first decompose the given graph into trails, and then we use a disentangling lemma, that allows us to transform this decomposition into one consisting only of paths. With this approach, we tackle three conjectures on H-decomposition of graphs and obtain results for the case H = P` is the path of length `. Two of these results solve weakenings of a conjecture of Kouider and Lonc (1999) and a conjecture of Favaron, Genest and Kouider (2010), both for regular graphs. We prove that, for every positive integer `, (i) there is a positive integer m0 such that, if G is a 2m`-regular graph with m ≥ m0, then G admits a P`-decomposition; (ii) if ` is odd, there is a positive integer m0 such that, if G is an m`-regular graph with m ≥ m0 and containing an m-factor, then G admits a P`-decomposition. The third result concerns highly edge-connected graphs: there is a positive integer k` such that if G is a k`-edge-connected graph whose number of edges is divisible by `, then G admits a P`-decomposition. This result verifies for paths the Decomposition Conjecture of Barát and Thomassen (2006), on trees. This work is an extended abstract of the Ph.D. thesis of the first author, written under the supervision of the second author.of the Ph.D. thesis of the first author, written under the supervision of the second author.