A　k-bisection　of　a　graph　G　is　a　partition　of　its　vertex　set　into　two　parts　V1　and　V2　of　the　same　cardinality　such　that　every　connected　component　of　the　subgraph　of　G　induced　by　Vi　(i=1,2)　has　at　most　k　vertices.　Ban　and　Linial　conjectured　that　every　bridgeless　cubic　simple　graph　admits　a　2-bisection　except　for　the　Petersen　graph.　Recently,　Abreu　et　al.　showed　that　Ban–Linial＇s　conjecture　is　true　for　bridgeless　claw-free　cubic　simple　graphs.　In　this　short　note,　we　prove　a　stronger　result.　We　show　that　for　any　(not　necessarily　bridgeless)　claw-free　cubic　multigraph　G　and　for　any　edge　e∈E(G),　there　exists　a　2-bisection　of　G　such　that　the　two　endvertices　of　e　belong　to　different　parts.　©　2018　Elsevier　B.V.