Decomposing 8-regular graphs into paths of length 4

A T-decomposition of a graph G is a set of edge-disjoint copies of T in G that cover the edge set of G. Graham and Häggkvist (1989) conjectured that any 2ℓ-regular graph G admits a T-decomposition if T is a tree with ℓ edges. Kouider and Lonc (1999) conjectured that, in the special case where T is the path with ℓ edges, G admits a T-decomposition D where every vertex of G is the end-vertex of exactly two paths of D, and proved that this statement holds when G has girth at least (ℓ+3)∕2. In this paper we verify Kouider and Lonc’s Conjecture for paths of length 4.