Decompositions of graphs into trees, forests, and regular subgraphs

A conjecture by A. Hoffmann-Ostenhof suggests that any connected cubic graph G contains a spanning tree T for which every component of G-E(T) is a K₁, a K₂, or a cycle. We show that any cubic graph G contains a spanning forest F for which every component of G-E(F) is a K₂ or a cycle, and that any connected graph G ≠ K₁ with maximal degree at most 3 contains a spanning forest F without isolated vertices for which every component of G-E(F) is a K₁, a K₂ or a cycle. We also prove a related statement about path-factorizations of graphs with maximal degree 3.