{"title":"Periodic Steiner Networks Minimizing Length","authors":"Jerome Alex, Karsten Grosse-Brauckmann","doi":"10.1007/s00454-023-00576-z","DOIUrl":null,"url":null,"abstract":"Abstract We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain with the same volume. The minimizer is the so-called network with quotient the complete graph on four vertices $$K_4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> . For comparison we consider a competing topological class, also with a quotient on four vertices, and determine the minimizing networks.","PeriodicalId":356162,"journal":{"name":"Discrete and Computational Geometry","volume":"42 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete and Computational Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00454-023-00576-z","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 7
Abstract
Abstract We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain with the same volume. The minimizer is the so-called network with quotient the complete graph on four vertices $$K_4$$ K4 . For comparison we consider a competing topological class, also with a quotient on four vertices, and determine the minimizing networks.
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