A　bisection　of　a　graph　is　a　bipartition　of　its　vertex　set　in　which　the　number　of　vertices　in　the　two　parts　differ　by　at　most　1,　and　its　size　is　the　number　of　edges　which　go　across　the　two　parts.　Let　Ck　be　a　cycle　of　length　k,　and　let　G　be　a　C4-free　graph　with　n　vertices,　m　edges　and　vertex　degrees　d1,…,dn.　Lin　and　Zeng　proved　that　if　G　does　not　contain　C6　and　has　a　perfect　matching,　then　G　admits　a　bisection　of　size　at　least　m/2+Ω(∑i=1ndi).　This　extends　a　celebrated　bound　given　by　Shearer　on　Max-Cut　of　triangle-free　graphs.　In　this　paper,　we　establish　a　similar　result　by　replacing　C6　with　θ(1,2,4),　θ(2,3,3)　and　θ(3,3,3),　where　θ(ℓ1,ℓ2,ℓ3)　denotes　the　graph　consisting　of　three　internally　disjoint　paths　of　length　ℓ1,　ℓ2　and　ℓ3,　respectively,　each　with　the　same　endpoints.　We　also　note　that　the　bound　is　tight　for　certain　polarity　graphs.　©　2022　Elsevier　B.V.