It　was　conjectured　by　Hoffmann-Ostenhof　that　the　edge　set　of　every　connected　cubic　graph　can　be　decomposed　into　a　spanning　tree,　a　matching　and　a　family　of　disjoint　cycles.　In　this　paper,　we　show　that　the　conjecture　is　true　for　connected　claw-free　cubic　graphs,　and,　furthermore,　any　edge　not　contained　any　triangle　appears　only　on　the　tree　or　on　the　matching.　Then　we　show　that　the　edge　set　of　every　connected　cubic　graph　(except　for　K-4　and　K-3,K-3)　can　be　decomposed　into　a　spanning　tree　and　a　family　of　disjoint　paths　of　length　at　most　2.　(C)　2020　Elsevier　B.V.　All　rights　reserved.