Decompositions of graphs into trees, forests, and regular subgraphs

A conjecture by A. Hoffmann-Ostenhof suggests that any connected cubic graph GG contains a spanning tree TT for which every component of G−E(T)G−E(T) is a K1K1, a K2K2, or a cycle. We show that any cubic graph GG contains a spanning forest FF for which every component of G−E(F)G−E(F) is a K2K2 or a cycle, and that any connected graph G≠K1G≠K1 with maximal degree at most 3 contains a spanning forest FF without isolated vertices for which every component of G−E(F)G−E(F) is a K1K1, a K2K2 or a cycle. We also prove a related statement about path-factorizations of graphs with maximal degree 3.