Two　edge-disjoint　spanning　trees　of　a　graph　G　are　completely　independent　if　the　two　paths　connecting　any　two　vertices　of　G　in　the　two　trees　are　internally　disjoint.　It　has　been　asked　whether　sufficient　conditions　for　hamiltonian　graphs　are　also　sufficient　for　the　existence　of　two　completely　independent　spanning　trees　(CISTs).　We　prove　that　it　is　true　for　the　classical　Ore-condition.　That　is,　if　G　is　a　graph　on　n　vertices　in　which　each　pair　of　non-adjacent　vertices　have　degree-sum　at　least　n,　then　G　has　two　CISTs.　It　is　known　that　the　line　graph　of　every　4-edge　connected　graph　is　hamiltonian.　We　prove　that　this　is　also　true　for　CISTs:　the　line　graph　of　every　4-edge　connected　graph　has　two　CISTs.　Thomassen　conjectured　that　every　4-connected　line　graph　is　hamiltonian.　Unfortunately,　being　4-connected　is　not　enough　for　the　existence　of　two　CISTs　in　line　graphs.　We　prove　that　there　are　infinitely　many　4-connected　line　graphs　that　do　not　have　two　CISTs.　(C)　2014　Elsevier　B.V.　All　rights　reserved.