The　Maximum　Core　Spanning　Tree　(MCST)　is　a　representative　cohesive　structure　generated　based　on　k-core,　which　is　the　maximum　edge　weight　spanning　tree　indicating　the　＇staired　coreness　hierarchy＇　in　each　connected　component.　The　edge　weight　here　is　defined　as　wuv=min{core(u),core(v)},　and　core(x)　is　the　corness　of　vertex　x.　Unlike　the　insertion　maintenance　problem　of　Maximum　Spanning　Tree　(MST)　which　has　known　efficient　algorithms,　MCST　insertion　maintenance　raises　special　challenges,　which　is　mainly　due　to　the　cascaded　vertex　coreness　changes　after　single-edge　insertion.　In　this　paper,　we　show　a　series　of　properties　of　MCST　and　MCST　insertion　maintenance　problem　and　propose　a　LoopFree　algorithm　to　maintain　the　MCST　efficiently.　In　particular,　the　time　complexity　for　MCST　maintenance　for　edge　insertion　is　bounded　by　O(|E∗|+|V|),　where　E∗　is　the　edge　set　whose　edge　weight　changes　after　insertion.　Through　extensive　evaluations,　we　show　the　proposed　MCST　insertion　maintenance　algorithm　has　good　efficiency　on　real-world　datasets.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.　2024.