A　bisection　of　a　graph　G　is　a　partition　of　its　vertex　set　into　two　sets　which　differ　in　size　by　at　most　1,　and　its　size　is　the　number　of　edges　between　the　two　sets.　The　Max-Bisection　problem　is　to　find　a　bisection　of　G　maximizing　its　size.　An　(l,r)-fan　is　a　graph　on　(r−1)l+1　vertices　consisting　of　l　cliques　of　order　r　that　intersect　in　exactly　one　common　vertex.　Let　G　be　a　connected　graph　with　n　vertices,　m　edges,　and　minimum　degree　at　least　2.　In　this　paper,　we　show　that　if　G　contains　neither　K2,l　nor　(l,4)-fan,　then　G　has　a　bisection　of　size　at　least　m∕2+(n−l+1)∕4,　which　improves　the　result　given　by　Jin　and　Xu.　As　a　corollary,　it　implies　that　every　connected　graph　with　n　vertices,　m　edges,　and　minimum　degree　at　least　2　and　without　cycles　of　length　4　has　a　bisection　of　size　at　least　m∕2+(n−1)∕4.　This　improves　a　result　given　by　Fan,　Hou　and　Yu.　©　2019　Elsevier　B.V.