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GlobalJournalofPureandAppliedMathematics,ISSN0793 1768,OnlineISSN0793 9750,Vol.9,№1,2013,pp.1 11AWrittenProofoftheFour ColorsMapProblemZhangTianshuNanhaiwestoilcorporation,ChinaoffshorePetroleum,Zhanjiangcity,Guangdongprovince,P.R.China.Email:xinshijizhang@hotmail.comAbstractAcontactborderoftwoadjacentfigurescanonlybetwoadjacentborderlines.Letusconsidertheplaneofanyuncoloredplanarmapaswhichconsistsoftwokinds’parallelstraightlinearsegmentsaccordingtoastripofakindalternatingastripofanother,andeverystraightlinearsegmentofeachkindconsistsoftwokindsofcoloredpointsaccordingtoacoloredpointofakindalternatingacoloredpointofanother,eitherkindofcoloredpointsatastraightlinearsegmentisnotaliketoeitherkindofcoloredpointsateitheradjacentstraightlinearsegmentofthestraightlinearsegment.Anyhowtheplanehasaltogetherfourkindsofcoloredpoints.Attheoutset,weneedtransformandclassifyfiguresatanuncoloredplanarmap.Firstmergeorderlyeachfigurewhichadjoinsatmostthreefiguresandanadjacentfigurewhichadjoinsatleastfourfiguresintoafigure.Secondlymergeeachtractoffigureswhichadjoinatmostthreefiguresandanadjacentfigureintoafigure.Afterthat,transformeveryborderlineclosedcurveoffigureswhichcomposedirectlythemergingfigureintotheframeofarectanglewhichhasonlylongitudinalandtransversalsides,accordingtothesequencefromoutsidemergingfiguretoinsidemergingfigure.Finallycoloreachfigurewithacoloraccordingtoeitheracolorofsomeparticularpointsofarectangularborderlinesclosedcurveofthefigure,oracolorunlikecolorsofitsadjacentfigures.KeywordsPlanarmap,figures,rectangles,fourcolors,writtenproof,borderlineclosedcurve,coloredpoints,classify,topologicaltransformation.BasicConceptsThefour colorsmapproblemstatesthatcoloreachandeveryfigurewithacoloratanyplanarorsphericalmapinsuchawaythattwoadjacent1figuresarecoloreddifferently,thenfourcolorsaresuffice.Weconsiderfiguresatsphericalmapassuchfigureswhichprintattherubberfilm,thensolongaswecutaholeinsideanyfigure,andtheholecont
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