Combinatorial　optimization　(CO)　on　graphs　is　a　classic　topic　that　has　been　extensively　studied　across　many　scientific　and　industrial　fields.　Recently,　solving　CO　problems　on　graphs　through　learning　methods　has　attracted　great　attention.　Advanced　deep　learning　methods,　e.g.,　graph　neural　networks　(GNNs),　have　been　used　to　effectively　assist　the　process　of　solving　COs.　However,　current　frameworks　based　on　GNNs　are　mainly　designed　for　certain　CO　problems,　thereby　failing　to　consider　their　transferable　and　generalizable　abilities　among　different　COs　on　graphs.　Moreover,　simply　using　original　graphs　to　model　COs　only　captures　the　direct　correlations　among　objects,　which　does　not　consider　the　mathematical　logicality　and　properties　of　COs.　In　this　paper,　we　propose　a　unified　pre-training　and　adaptation　framework　for　COs　on　graphs　with　the　help　of　the　maximum　satisfiability　(Max-SAT)　problem.　We　first　use　Max-SAT　to　bridge　different　COs　on　graphs　since　they　can　be　converted　to　Max-SAT　problems　represented　by　standard　formulas　and　clauses　with　logical　information.　Then　we　further　design　a　pre-training　and　domain　adaptation　framework　to　extract　the　transferable　and　generalizable　features　so　that　different　COs　can　benefit　from　them.　In　the　pre-training　stage,　Max-SAT　instances　are　generated　to　initialize　the　parameters　of　the　model.　In　the　fine-tuning　stage,　instances　from　CO　and　Max-SAT　problems　are　used　for　adaptation　so　that　the　transferable　ability　can　be　further　improved.　Numerical　experiments　on　several　datasets　show　that　features　extracted　by　our　framework　exhibit　superior　transferability　and　Max-SAT　can　boost　the　ability　to　solve　COs　on　graphs.