{"title":"The Clique-Rank of 3-Chromatic Perfect Graphs","authors":"J. Fonlupt","doi":"10.1137/1.9780898718805.ch5","DOIUrl":null,"url":null,"abstract":"The clique-rank of a perfect graph G introduced by Fonlupt and Sebö is the linear rank of the incidence matrix of the maximum cliques of G. We study this rank for 3-chromatic perfect graphs. We prove that if, in addition, G is diamond-free, G has two distinct colorations. An immediate consequence is that the Strong Perfect Graph Conjecture holds for diamond-free graphs and for graphs with clique number equal to three. The proofs use both linear algebra and combinatorial arguments.","PeriodicalId":416196,"journal":{"name":"The Sharpest Cut","volume":"10 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Sharpest Cut","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/1.9780898718805.ch5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The clique-rank of a perfect graph G introduced by Fonlupt and Sebö is the linear rank of the incidence matrix of the maximum cliques of G. We study this rank for 3-chromatic perfect graphs. We prove that if, in addition, G is diamond-free, G has two distinct colorations. An immediate consequence is that the Strong Perfect Graph Conjecture holds for diamond-free graphs and for graphs with clique number equal to three. The proofs use both linear algebra and combinatorial arguments.
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