A　spanning　subgraph　F　of　a　graph　G　is　called　an　even　factor　of　G　if　each　vertex　of　F　has　even　degree　at　least　2　in　F.　It　was　conjectured　that　if　a　graph　G　has　an　even　factor,　then　it　has　an　even　factor　F　with　,　where　is　the　set　of　vertices　of　degree　2　in　G.　We　note　that　the　conjecture　is　false　if　G　is　a　triangle.　In　this　paper,　we　confirm　the　conjecture　for　all　graphs　on　at　least　4　vertices,　and　moreover,　we　prove　that　if　for　every　even　factor　H　of　G,　then　every　maximum　even　factor　of　G　is　a　2-factor　consisting　of　even　circuits.