关于若干整数分拆问题

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1、万方数据第26卷第l期苏州科技学院学报(自然科学版)V01．26No．120a19年3月JournalofSuzhouUniversityofScienceandTechnology(NaturalScience)Mar．2()09ProblemsonIntegerDecompositionZHOUJian-qinl，一，ZANGYu-bingz(1．TelecommunicationSchool，Hangzhou1)iamiUniversity，Hangzhou310018。China；2．DepartmentofComputerScience，AnhuiUniversity0fTechn

2、ology，Ma'anshan243002，China；3．YanzhouCollege，ShandongTVUniversity，Yanzhou272100，China)Abstract：Y．Alavi，Aj．Boats，G．Chartrand，P．Erd酗and0．R．OellermannhavepDp08edthefolhwingconjecture：Letat，如，⋯。mbeintegerswhich舶ti8母n《啦≤2n—

3、2，l《i≤蠡，andat+arl-⋯+q2n(n+1)／2．ThenS={1，2，⋯，埘containskdisjointsubsetsSI,是，⋯，Swhi

4、ch醴tis匆f(1≤i≤蠡)，僻=暑(S)．Asanaturalgeneralizationoftheconjecture，Wehaveobtainedthefollowingtheorem：Let口l，吨，⋯，皿beintegerswhich蛆ti由嘎≥n，1≤i≤矗，andat+a2．+⋯+峨≤厅(厅+1)尼．Then3=11，2，⋯，mcontainskdisjointsubsetsSl，52，⋯，s．thatsatisfyi(1《i≤知)andq=乏(sj)．Withthetheorem,wehaveprovedthemaintheoremobtainedbyICAndo，S．Ge

5、rvadoandM．Kano．Atthesa船time，锄errorinthebibliogra-phyiscorrected．Keywords：integerdecomposition；trialgraph；functioncKn雠LibraryCl脚mtflcaflon：0174．2MR(2000)SubjectClassification：05A17；05C70；11875；90C35LiteratureLabelCode：APaperNumber：1672-0687(2009)01-0006-051IntroductionFordefinitions，notations，algori

6、thm,lemmasandtheoremsnotpresentedhere，wefollow【1】．Forconve-nience，wensethefollo丽ngnotation∑∞)：艺xXE^Letat，屹，⋯，atbeintegers，then(nl，az，⋯，啦lissaidtobedecomposedon{1，2，⋯，厅lif(1，2，⋯，nlcontainskdisjointsubsetsS1，岛，⋯，＆suchthatforeachi(1≤f≤

7、

8、})，驴乞．『．WecallSt鹊thedecompositionofjEsInf(1≤i≤后)，and{口l，锄，⋯，m】iss

9、aidtobedecomposableon{l，2，⋯，n}．Y．Alavi，A．J．Boats，G．Chartrand，P．Erdi，sand0．R．Oellermannhaveproposedthefollowingconjecturein【2】：ConjectureALetal，a2，⋯，诹beintegerssuchthatn≤哦≤2尼一2，l≤i≤

10、

11、}，andm+口2+⋯+at=n(n+1)／2．Then{口l，n2，⋯，啦lisdecomposableonIl，2，⋯，，11．K．Ma,HIzhouandJ．Zhouin【l】prop08edadecompositionalgo

12、rithm，thenprovedConjectureAbyprovingthevalidityofthedecompositionalgorithm．Wegeneralizethedecompositionalgorithmin【1】．Byprovingthevalidityofthegeneralizeddecompositionalgorithm，weprovethefollowingtheorem：Th