Higher edge-connectivities and tree decompositions in regular graphs

A tree decomposition of a graph G is a family of subtrees whose edge sets partition the edge set of G. The arboricity, a(G), is a trivial lower bound of τ(G), the minimum number of trees in a tree decomposition. We prove that a(G)=τ(G) for all regular graphs of order n and degree d⩾⌊n/2⌋. This bound is best possible. The proof uses higher edge-connectivities, which also provide sufficient conditions for a(G)=τ(G) when d>2n.